Principles and Practice of Engineering PE Chemical Reference Handbook by National Council of Examiners for Engineering and Surveying (NCEES) (z-lib.org)

Chapter 1: General Information

1.3.2.3 Cubic Geometry—Volume and Surface Area
A = surface area
V = volume
a,b,c = lengths of sides
d = diagonal(s) or diameter
h = height

Cube c d

V = a3
A = 6a2
d=a 3

Cuboid c a b
d b
V = abc
A = 2 (a b + a c + b c) a A base
d = a2 + b2 + c2

Parallelepiped
V = Abase h

h

Pyramid
Abase h
V = 3

h

A base

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Chapter 1: General Information

Frustum of Pyramid

V = h _ A1 + A2 + A1 A2 i A2
3 A1
h
A1 + A2
V . h 2 for A1 . A2

Right Circular Cylinder

V = r d2 h r
4 h

Amantle = 2 r r h d

A = 2 r r (r + h) h
d
Hollow Cylinder
r
V = 4 h (D2 − d2)

Right Circular Cone D A2
r r m
V = 3 r2 h = 12 d2 h x
h A1
Amantle = r r m
r —m2 m
A = r r (r + m) p
hd
m = h2 + r2 d

A1: A2 = x2: h2

Frustum of Cone

V = r h (D2 + D d + d2)
12

Amantle = r m (D + d) = 2r pm
2

m= h2 + c D − d 2
2
m

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Chapter 1: General Information

Sphere

=V 34=r r3 1 r d3
6

=A 4=r r2 r d2

Zone of a Sphere

V = r h (3a2 + 3b2 + h2) a
6 r

Amantle = 2 r r h h
b
A = r (2 r h + a2 + b2)

Segment of a Sphere (Spherical Cap)

V = r h c 3 s2 + h2m = r h2cr − h m h
6 4 3 sr

Acap = 2r r h = r (s2 + 4 h2)
4

Acap = 2 r r2 (1 − cos i0)

s = 2 h (2 r − h)

q0 is the angle of the cutout, rotated from the center
of the radius.

Sector of a Sphere

V = 2 r r2 h r
3

A = r r (4 h + s) h
2 s

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Chapter 1: General Information

Sphere with Cylindrical Boring

V = r h3
6

A = 2 r h (R + r)

h = R2 − r2 h R

r

Sphere with Conical Boring

V = 2 r R2 h
3

A = 2r h (R + D ) h R
2

D = 2 R2 − h2

Torus D
D
V = r2 D d2
4
d
A = r2 D d

Sliced Cylinder
r
V = 4 d2 h

h

Ungula d
h
V = 2 r2 h
3 r

Amantle = 2 r h

A = r2 >2 h + r e1 + 1 + h2 oH
r 2 r2

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Chapter 1: General Information

Barrel
r
V . 12 h (2 D2 + d2)

h D

Prismoid d

V = h (A1 + A2 + 4 A) h A2
6 h/2 A

A1

Regular Polyhedra No. of Faces Form of Faces Total Surface Area Volume
4 Equilateral triangle 1.7321a2 0.1179 a3
Name 6 Square 6 a2
Tetrahedron 8 Equilateral triangle 3.4641a2 a3
Cube 12 Regular pentagon 20.6457 a2
Octahedron 20 Equilateral Triangle 8.6603 a2 0.4714 a3
Dodecahedron 7.6631a3
Icosahedron d 2.1817 a3

The radius of a sphere inscribed within a regular polyhedron is:

r = 3V
A

Paraboloid of Revolution
r
V = 8 h d2

h

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Chapter 1: General Information

1.3.3 Calculus

1.3.3.1 Differentiation dy
dx
For any function y = f(x), the derivat=ive D=x y = yl

yl = limit d Dy n
Dx
Dx " 0

= limit * 8 f_x + Dxi - f^xhB 4
^Dxh
Dx " 0

where yl = the slope of the curve f^xh

Test for a Maximum
y = f^xh is a maximum for x = a, if f l^ah = 0 and f m^ah 1 0

Test for Minimum
y = f^xh is a minimum for x = a, if f l^ah = 0 and f m^ah 2 0

Test for a Point of Inflection

y = f(x) has a point of inflection at x = a, if f m^ah = 0, and if f m^xh changes sign as x increases
through x = a

L'Hôpital's Rule f^xh 3
g^xh 3
If the fractional function assumes one of the indeterminate forms 0 or (where a is finite or infinite), then:
0

limit f^xh
g^xh
x"a

is equal to the first of the expressions

limit f l^xh limit f m^xh limit f n^xh
gl^xh gm^xh gn^xh
x"a x"a x"a

which is not indeterminate, provided such first indicated limit exists.

Curvature K of a Function Y y = f (x)
Q
The curvature K of a curve at point P is the limit of its average α
curvature for the arc PQ as Q approaches P. This is also O Δ s =Δα
expressed as: P

The curvature of a curve at a given point is the α + Δα
rate of change of its inclination with respect to its arc length. X

=K lD=ims "i0t DDas da
ds

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Chapter 1: General Information

Curvature in Rectangular Coordinates

K= ym

3
91 + _ yli2C2

When it may be easier to differentiate the function with respect to y rather than x, the notation xl will be used for the derivative.

xl = dx
dy
−xm
K=
3
81 + ^xlh2B2

Radius of Curvature

The radius of curvature R at any point on a curve is defined as the absolute value of the reciprocal of the curvature K at that
point.

R= 1 _K ! 0i
K

3 _ym ! 0i

R = 91 + _ yli2C2
ym

List of Derivatives
u, v, and w represent functions of x.

a, c, and n represent constants.

Arguments of trigonometric functions are in radians. The following definitions are used:

arcsin u = sin-1 (u), ^sin uh−1 = 1 u
sin

1. dc = 0
dx

2. dx = 1
dx

3. d^cuh = c du
dx dx

4. d_u +v − wi = du + dv − dw
dx dx dx dx

5. d^uvh = u dv + v du
dx dx dx

6. d ^u v wh = u v dw + u w dv + v w du
dx dx dx dx

dc u m v du −u dv
v dx v2 dx
7. =
dx

8. d_uni = n un − 1 du
dx dx

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Chapter 1: General Information

9. d7 f ^uhA = ) d7 f ^uhA 3 du
dx du dx

10. du = 1
dx
c dx m
du

11. d_loga u i = _loga e i 1 du
dx u dx

12. d^ln uh = 1 du
dx u dx

13. d_aui = ^ln ahau du
dx dx

14. d_eui = eu du
dx dx

15. d_uvi = vuv−1 du + ^ln uhuv dv
dx dx dx

16. d^sin uh = cos u du
dx dx

17. d^cos uh = − sin u du
dx dx

18. d^tan uh = sec2u du
dx dx

19. d^cot uh = − csc2u du
dx dx

20. d^sec uh = sec u tan u du
dx dx

21. d^csc uh = − csc u cot u du
dx dx

22. d_sin−1ui = 1 u2 du c -r # sin-1u # r m
dx 1− dx 2 2

23. d_cos−1ui =− 1 u2 du _0 # cos-1u # ri
dx 1− dx

24. d_tan−1ui = 1 1 du c -r 1 tan-1u 1 r m
dx + u2 dx 2 2

25. d_cot−1ui = − 1 1 du _0 1 cot-1u 1 ri
dx + u2 dx

26. d_sec−1ui = u 1 − 1 du c0 1 sec-1u # r m and c-r # sec-1u - r m
dx u2 dx 2 2

27. d_csc−1ui = − u 1 − 1 du c0 1 csc-1u # r m and c-r 1 csc-1u # - r m
dx u2 dx 2 2

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Chapter 1: General Information

Parametric Form of the Derivative

yl_x^ t=hi dd=yx dy dt = yo
dt dx xo

ym_x^ t hi = d2y = xo yp − yo xp
^dxh2 xp

where

yo = dy
dt

yp = d2y
dt 2

Derivative of Inverse Functions
The equation y = f(x) solved for x gives the inverse function x = {_ yi.

f l^ xh = 1 yi
{l _

1.3.3.2 Integration

The indefinite integral F(x) is a function such that Fl]xg = f]xg .

# f^xhdx = F^xh + C

C is an unknown constant which disappears on differentiation.

The definite integral:

/ #n b b
limit f_ xi iDxi = f^xhdx = F (x) a = F^bh − F^ah

n"3 i=1 a

Also, Dxi " 0 for all i.

To find the integral: Use the list of indefinite integrals (below), integration by parts (equation #6 in the list), integration by substi-
tution, and separation of rational fractions into partial fractions.

List of Indefinite Integrals
u, v, and w represent functions of x.

a, c, and n represent constants.

Arguments of trigonometric functions are in radians. The following definitions are used:

arc sin u = sin−1^uh, ^sin uh−1 = 1
sin u

Note: A constant of integration should be added to the integrals.

#1. d f^xh = f^xh
#2. dx = x
# #3. a f^xhdx = a f^xhdx
# # #4. 7u^xh ! v^xhAdx = u^xhdx ! v^xhdx

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Chapter 1: General Information

#5. xmdx = xm+1 _m !- 1i
m+1

# #6. u^xhdv^xh = u^xhv^xh - v^xhdu^xh

#7. dx = 1 ln ax + b for a = 1 and b = 0: # dx = ln x
ax + b a x

#8. dx =2 x
x
#9. ax
a x dx = ln a

#10. sin x dx = - cos x

#11. cos x dx = sin x

#12. sin2x dx = x - sin 2x
2 4

#13. cos2x dx = x + sin 2x
2 4

#14. x sin x dx = sin x - x cos x

#15. x cos x dx = cos x + x sin x

#16. sin x cos x dx = sin2x
2

#17. sin a x cos b x dx = - cos_a − bix - cos_a + bix _a2 ! b2i
2_a − bi 2_a + bi
#18. =tan x dx -=ln cos x ln sec x ^x 2 0h
_a ! 0i
#19. =cot x dx -=ln csc x ln sin x _a 2 0, c 2 0i
_4ac - b2 2 0i
#20. tan2x dx = tan x - x _b2 - 4ac 2 0i
_b2 - 4ac = 0i
#21. cot2x dx = - cot x − x

#22. eaxdx = c 1 meax
a

#23. xeaxdx = e eax o^ax - 1h
a2

#24. ln x dx = x8ln^xh − 1B

#25. dx = 1 tan−1 x
a2 + x2 a a

#26. dx c = 1 tan−1 c x a m
ax2 + ac c

#27a. ax2 dx + c = 2 b2 tan−1 2ax +b
+ bx 4ac − 4ac − b2

#27b. ax2 dx + c = 1 ln 2ax + b - b2 - 4ac
+ bx b2 − 4ac 2ax + b + b2 - 4ac

27c. # dx +c = - 2 b
ax2 + bx 2ax +

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Chapter 1: General Information

1.3.3.3 Multivariable Calculus

Partial Derivatives

In a function of two independent variables x and y, a derivative with respect to one of the variables may be found if the other variable is as-
sumed to remain constant. If y is kept fixed, the function

z = f_x, yi

becomes a function of the single variable x, and its derivative (if it exists) can be found. This derivative is called the partial derivative of

z with respect to x. The partial derivative with respect to x is denoted as follows:

2z = 2f (x, y)
2x 2x

Total Derivative

Given f(x,y), then the total derivative df is

df = d 2f n dx + e 2f o dy
2x 2y
y x

Chain Rule

Given f_x, yi where x = g^ t h and y = h^ t h, then

df = d 2f n dx + e 2f o dy
dt 2x dt 2y dt
y x

Identities in Partial Derivatives
2x
c 2x m = 1

z

c 2x m = 0
2z
x

Implicit Differentiation
If f (x,y,z) cannot be converted to an explicit expression in the form of z = f )_x, yi, then

2z −d 2f n 2z −e 2f o
2x 2x 2y 2y
c m = y,z and d n = x,z

y d 2f n x d 2f n
2z 2z
x, y x,y

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Chapter 1: General Information

Rules for changing the constant or the variable on a partial derivative:

Given f (x,y,z) = constant, then

d 2f n = d 2f n + e 2f od 2y n
2x 2x 2y 2x
z y x z

d 2f n = e 2f o d 2y n
2z 2y 2z
x x x

Cyclic relation:

c 2z m : d 2x n : d 2y n =− 1
2x 2y 2z
y z x

Reciprocity relation:

c 2z m = 1
2x
y c 2x m
2z
y

1.3.3.4 Differential Equations

A common class of ordinary linear differential equations is

bn d n y^xh + ... + b1 dy^xh + b0 y^xh = f^xh
dx n dx

where bn, ... , bi, ... , b1, b0 are constants.

When the equation is a homogeneous differential equation, f(x) = 0, the solution is

y h ^xh = C1 e r1 x + C 2 e r2 x + . . . + Ci e ri x + C n e rn x

where rn is the nth distinct root of the characteristic polynomial P(x) with
P^rh = b n r n + b n − 1 r n − 1 + b1 r + b0

If the root r1 = r2, then C2 e r2 x is replaced with C2 xe r1 x .

Higher orders of multiplicity imply higher powers of x. The complete solution for the differential equation is

y(x) = yh(x) + yp(x)
where yp(x) is any particular solution with f(x) present. If f(x) has ernx terms, then resonance is manifested. Furthermore, specific
f(x) forms result in specific yp(x) forms, some of which are

f(x) yp(x)
A B
Aeax
Beax, a ! rn
A1 sin ~x + A2 cos ~x
B1 sin ~x + B2 cos ~x

Common First-Order Differential Equations and Their Solutions

Form Solution Substitution/Conditions
Linear, homogeneous ODE y (x) = C e−ax
with constant coefficients y (x) = C e(−# p(x)dx) C is a constant that satisfies the
initial condition.
yl + a y = 0
Linear, homogeneous ODE C is a constant that satisfies the
initial condition.
yl + p^xh y = 0

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Chapter 1: General Information

Common First-Order Differential Equations and Their Solutions (cont'd)

Form Solution Substitution/Conditions

Linear, inhomogeneous ODE y (t) = KA + (KB − KA)a1 =− e− xt k p (t) '=BA tt 12 00 1 y (0) K A
with constant coefficients
t = ln e KB − KA o =x ti=me constant, K gain
x yl + y = K p (t) x KB − y

Comment: Solution is for a step function.

Implicit ODE, no y term x = f (p) Substitution:
x = f_ yli y = # p f (p) dp + C yl = p

Comment: Elimination of p leads to a solution in parametric form.

x= # f (p) dp + C
p
Implicit ODE, no x term yl = p
y = f_ yli y = f (p)

Comment: Elimination of p leads to a solution in parametric form.

Separable ODE # g_ yidy = # f^xhdx + C

=yl dd=xy f^xh
g_ yi
Comment: The variables x and y can be separated into the left and right

sides of the equation.

# # Substitution: u = y
x
dx = du +
Similarity ODE x f^uh − u C yl = u + x du
dx
yl = fc y m
x
y
Comment: Check whether it is possible to transform to f c x m.

Common Second-Order Differential Equations and Their Solutions

Form Solution Substitution

ODE, y and y' terms missing # #y(x) = C1 + C2 x + ; f^xhdxEdx

ym = f^xh Comment: Start the calculation with the inner integral.

ODE, y term missing # #du = − p1^xhdx + C1 Substitution: u = y'
ym + p1^xh f_ yli = 0 f^uh =ym dd=ux f_ yli f^uh
#y = udx + C2

du Substitution: u = yl
dy
ODE, x term missing u = f_ y, ui =ym dd=ux u du = f (y, u)
ym = f(y, yl) dy
dy
x = # u_ yi + C Then substitute yl = dy for u.
dx

=where u u=(y) and y y (x)

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Chapter 1: General Information

Common Second-Order Differential Equations and Their Solutions

Form Solution Substitution

Solution depends on the values of r1,2 = 1 `− a ! a2 − 4b j
a and b. 2

Linear, homogeneous ODE y (x) = C1 er1x + C2 er2x a2 2 4b (overdamped)
with constant coefficients y (x) = (C1 + C2 x) er1x a2 = 4b (critically damped)

ym + a yl + b y = 0 y (x) = ea x

[C1 cos (b x) + C2 sin (b x)] a2 1 4b (underdamped)

a = − 1 a b = 1 4b − a2
2 2

1.3.3.5 The Fourier Transform and Its Inverse

#X_ f i = +3 x^the −j2rftdt
−3

#x^th = +3 X_ f ie j2rftdf
−3

We say that x(t) and X(f) form a Fourier transform pair:

x^t h * X_ f i

Fourier Transform Pairs

Fourier Transform Pairs
x(t) X(f)
1 d_ f i

d^t h 1

u^t h 1 d_ f i + 1
2 j2rf

Pc t m x sinc _xf i
x

sinc ^Bth 1 Pd f n
B B

Kc t m x sinc 2_xf i
x

e-atu^ t h 1 a20
te-atu^ t h a + j2rf

e-a t 2a a20
e-^ath2 a2 + _2rf i2

2a a20
a2 + _2rf i2

r e-c rf 2
a a
m

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Chapter 1: General Information

Fourier Transform Pairs (cont'd)
x(t) X(f)

cos`2rf0t + ij 1 9e ji d` f − f0 j + e−ji d` f + f0jC
2

sin`2rf0t + ij 1 9e ji d` f − f0j − e −ji d ` f + f0jC
2j

/n =+3 /k =+3 fs = 1
d_t − nTsi Ts
fs d` f − kfsj
n =−3
k =−3

Fourier Transform Theorems Fourier Transform Theorems

Linearity ax^ t h + by^ t h aX_ f i + bY_ f i
Scale change
Time reversal x^ath 1 Xc f m
Duality a a
Time shift
Frequency shift x_-ti X`-f j
Modulation
Multiplication X^t h x`-f j
Convolution x_t - t0j
Differentiation x^ t he-j2rf0t X_ f ie-j2rft0

Integration x^ t hcos 2rf0t X` f - f0j

x^t h: y^t h 1 X` f − f0j + 1 X` f + f0 j
x^t h* y^t h 2 2

dnx^t h X_ f i* Y_ f i
dtn
X_ f i: Y_ f i

_ j2rf inX_ f i

# t x(m)d m 1 X_ f i+ 1 X^0 h d _ f i
j2rf 2
-3

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Chapter 1: General Information

1.3.3.6 Laplace Transforms

The unilateral Laplace transform pair:

#F^sh = 3 f^ t he−st dt
0

#f^t h=1 v + j3 F^shest ds
2rj
v − j3

where s = s + jw

represents a powerful tool for the transient and frequency response of linear time invariant systems. Some useful Laplace
transform pairs are

Laplace Transform Pairs

f(t) F(s)
d(t), Impulse at t = 0 1

u(t), Step at t = 0 1
s

t[u(t)], Ramp at t = 0 1
s2

e-at 1
_s + ai

te-at 1
_s + ai2

e-at sin bt b
9_s + ai2 + b2C

e-at cos bt _s + ai
9_s + ai2 + b2C

d nf ^th /snF^sh −n−1 sn − m − 1 dm f ^0h
dt n m=0 dtm

# t f^xhdx c 1 mF^ s h
s
0
H^shX^sh
# t x_t - xih^xhdx
e- xs F^ s h
0

f_t - xiu_t - xi

tli"mi3t f^ t h limit sF^ sh
s"0

limit f^t h limit sF^ s h
t"0 s"3

The last two transforms represent the Final Value Theorem (F.V.T.) and Initial Value Theorem (I.V.T.), respectively. It is
assumed that the limits exist.

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Chapter 1: General Information

1.3.4 Statistics and Probability

1.3.4.1 Mean, Median, and Mode

If X1, X2, ... , Xn represents the values of a discrete random sample of n items or observations, the arithmetic mean of these items
or observations, denoted X , is defined as
/n
X = c 1 m_ X1 + X2 + ... + Xnj = c 1 m Xi
n n
i−1

X " n for sufficiently large values of n.

The weighted arithmetic mean is
X w = //wwi Xi i

where Xi = the value of the ith observation and wi = the weight applied to Xi.

The variance of the population is the arithmetic mean of the squared deviations from the population mean. If m is the arithme-
tic mean of a discrete population of size N, the population variance is defined by

v2 = c 1 m:_ X1 − ni2 + _ X2 − ni2 + ... + ` XN − nj2D
N

N
/= 1 _ Xi − nj2
c N m

i=1

Standard deviation formulas are

spopulation = c 1 m/_ Xi − nj2
N

ssum = v12 + v22 + ... + v2n

sseries = v n

smean = v
n

sproduct = A2 v2b + B2 va2

/The 1 n
−
sample variance is s2 = =_n 1iG _ Xi − X j2

i=1

/The sample standard deviation is = c n 1 1 m n _ Xi − X j2
− =
i 1

The sample coefficient of variation is CV = s
X

The sample geometric mean is n X1 X2X3 ...Xn

The sample root-mean-square value is c 1 m/Xi2
n
th
When the discrete data are rearranged in increasing order and n is odd, the median is the value of the c n + 1 item.
2 m

When n is even, the median is the average of the c n th and c n + th
2 2
m 1m items.

The mode of a set of data is the value that occurs with greatest frequency.

The sample range R is the largest sample value minus the smallest sample value.

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Chapter 1: General Information

1.3.4.2 Permutations and Combinations
A permutation is a particular sequence of a given set of objects. A combination is the set itself without reference to order.

The number of different permutations of n distinct objects taken r at a time is:

P_n, ri = _n n!
− ri!

An alternative notation for P(n,r) is nPr.

The number of different combinations of n distinct objects taken r at a time is:

C_n, ri = P_n, ri = n!
r! 8r!_n − ri!B

nCr and b n l are alternative notations for C(n,r).
r

The number of different permutations of n objects taken n at a time, given that ni are of type i, where i= 1, 2, ..., k and /ni = n,is

P_n; n1, n2, ..., nki = n!
n1!n2!...nk!

1.3.4.3 Probabilities

Property 1. General Character of Probability

The probability P(E) of an event E is a real number in the range of 0 to 1. The probability of an impossible event is 0 and that of
an event certain to occur is 1.

Property 2. Law of Total Probability
P^A + Bh = P^Ah + P^Bh − P_A, Bi

where P(A+B) = the probability that either A or B occurs alone or that both occur together

P(A) = the probability that A occurs

P(B) = the probability that B occurs

P(A,B) = the probability that both A and B occur simultaneously

Property 3. Law of Compound or Joint Probability
If neither P(A) nor P(B) is zero, then

P(A, B) = P(A) P(B | A) = P(B) P(A | B)
where

P(B | A) = the probability that B occurs given the fact that A has occurred
P(A | B) = the probability that A occurs given the fact that B has occurred
If either P(A) or P(B) is zero, then P(A, B) = 0.

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Chapter 1: General Information

Bayes' Theorem:

/P`B j | Aj =P`B jj P`A | B jj

n

P_A | Bii P_Bii

i=1

where P(Aj ) is the probability of event Aj within the population of A

P(Bj ) is the probability of event Bj within the population of B

1.3.4.4 Distributions and Expected Values

A random variable X has a probability associated with each of its possible values. The probability is termed a
discrete probability if X can assume only discrete values, or

X = x1, x2, x3, ..., xn
The discrete probability of any single event X = xi occurring is defined as P(xi) while the probability mass function of the ran-
dom variable X is defined by

f _xki = P_ X = xkj, k = 1, 2, ..., n

Probability Density Function
If X is continuous, the probability density function, f, is defined such that

#b

P^a # X # bh = f^xhdx

a

See the table of probability and density functions.

Cumulative Distribution Function
The cumulative distribution function, F, of a discrete random variable, X, that has a probability distribution
described by P(xi) is defined as

/m

=F_xmi =P_xki P_ X # xmi, m = 1, 2, ..., n

k=1

If X is continuous, the cumulative distribution function F is defined by

x

#F^xh = f^t hdt

−3

which implies that F(a) is the probability that X # a .

Expected Values
Let X be a discrete random variable having probability mass function:

f_xki, k = 1, 2, ..., n

The expected value of X is defined as

/n

=n E=6X @ xk f^xkh

k=1

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Chapter 1: General Information

The variance of X is defined as

/v2 = V6X @ = n _xk − nj2 f_xki
k=1

Let X be a continuous random variable having a density function f(X) and let Y = g(X) be some general function. The

expected value of Y is

3

#E6Y@ = E7g^X hA = g^xh f^xhdx

−3

The mean or expected value of the random variable X is now defined as

3

#n = E6X @ = xf^xhdx

−3

while the variance is

3

#v2 = V6X @ = E:_ X − ni2D = _x − ni2 f^xhdx

−3

The standard deviation is v = V6X @.

The coefficient of variation is defined as v .
n

Combinations of Random Variables
Y = a1 X1 + a2 X2 + ... + an Xn

The expected value of Y is ny = E^Y h = a1 E_ X1i + a2 E_ X2i + ... + an E_ Xni.

If the random variables are statistically independent, then the variance of Y is

v 2 = V^Y h = a12V_ X1i + a22V_ X2i + ... + an2V_ Xni
y

= a12 v12 + a22 v22 + ... + an2 vn2

Also, the standard deviation of Y is vy = v2y .

When Y = f(X1,X2,...,Xn) and Xi are independent, the standard deviation of Y is expressed as

vy = e 2f vx1 2 + e 2f vx2 2 + ... + e 2f vxn 2
2X1 2X2 2Xn
o o o

Normal Distribution (Gaussian Distribution)

This is a unimodal distribution, the mode being x = µ, with two points of inflection (each located at a distance σ to either side of

the mode). The averages of n observations tend to become normally distributed as n increases. The variate x is said to be normally

distributed if its density function f (x) is given by an expression of the form

f^xh = 1 − 1 d x − n 2
2r 2 v
e n

v

where

µ = the population mean

σ = the standard deviation of the population

-3 # x # 3

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Chapter 1: General Information

When µ = 0 and σ2 = σ = 1, the distribution is called a standardized or unit normal distribution. Then

f^xh = 1 e −x 2 /2
2r

where -3 # x # 3

It is noted that Z = x- n follows a standardized normal distribution function.
v

A unit normal distribution table is included in this section. In the table, the following notations are used:

F(x) = the area under the curve from –∞ to x

R(x) = the area under the curve from x to ∞

W(x) = the area under the curve between –x and x

F(-x) = 1 - F(x)

1.3.4.5 Confidence Intervals

Confidence Interval for the Mean n of a Normal Distribution

When standard deviation v is known:

X - Za/2 v # n # X + Za/2 v
n n

When standard deviation v is not known:

X - ta/2 s # n # X + ta/2 s
n n

where ta/2 corresponds to n - 1 degrees of freedom.

Confidence Interval for the Difference Between Two Means m1 and m2
When standard deviations s1 and s2 are known:

X1 - X2 - Za/2 v12 + v 2 # n1 - n2 # X1 - X2 + Za/2 v12 + v 2
n1 2 n1 2
n2 n2

When standard deviations s1 and s2 are not known:

c 1 + 1 m8_n1 - 1is12 + ^n2 - 1hs 22B
n1 n2 n1 + n2 - 2
X1 - X2 - ta/2 # n1 - n2 #

c 1 + 1 m8_n1 - 1is12 + ^n2 - 1hs 22B
n1 n2
X1 - X2 + ta/2
n1 + n2 − 2

where ta/2 corresponds to n1 + n2 - 2 degrees of freedom.

Confidence Intervals for the Variance v2 of a Normal Distribution

^n - 1hs2 # v2 # ^n - 1hs2
xa2/2,n - 1 x12- a/2,n - 1

Sample Size

z = X - n n = e v za/2 2
v xr −n
o

n

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Chapter 1: General Information

The Central Limit Theorem
Let X1, X2,...,Xn be a sequence of independent and identically distributed random variables having mean m and variance s2. Then
for a large n, the Central Limit Theorem asserts that the sum

Y = X1 + X2 + ... + Xn is approximately normal

n yr = n

and the standard deviation is vyr = v .
n

1.3.4.6 Probability and Density Functions

Probability and Density Functions

Kind of Probability Density Function f(x) Expected Mean (m), Form of the Density
Distribution Distribution Function F(x) Mean (x), Variance (s2) Function

General Comment: General distribution for continuous values
(continuous)
f (x) #x = 3
General
(discrete) #F(x) = x f(t)dt x f(x) dx
−3 −3
Uniform
#v2 = 3 x2 f (x) dx − n2
−3

Comment: General distribution for discrete values: n is the number in a random sample,

xi is the discrete value of the random variable, and Pi is the probability.

Pi /x = n 1 (xi Pi)
i=
/F (x) = i < x Pi /v2 =
n 1 (xi2 Pi − n2)
i=

Comment: Random variable x = 0 only within the interval <a, b>, where each value is of

equal probability. Use when only maximum and minimum values are known

but no other information about the distribution in between.

f (x) ==\]]]]]Z][]]]]]]]]]]\[Zbbx1−01−−0aaa for a # x # b x = a + b f(x)
F (x) 2
for outside 1
for − 3 1 x 1 a v2 = _b − ai2 b-a
12
for a # x # b oa μ b x
for b 1 x 1 3

Comment: If P(k) is the probability that in n random samples exactly k errors will occur,
the error probability is p. Lot size is assumed to be 3.

Binomial P(k) n = 20
0.3 p = 0.1
P (k) = c n m pk `1 − pjn − k x=np 0.2 p = 0.2p = 0.5
k v2 = n p (1 − p) 0.1

/F(x) = n pk `1 − pjn − k 0 5 10 15 k
x
k<x

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Chapter 1: General Information

Probability and Density Functions (cont'd)

Kind of Probability Density Function f(x) Expected Mean (m), Form of the Density
Distribution Distribution Function F(x) Mean (x), Variance (s2) Function

Normal Comment: Often obtained in practice as measured values with a bell-shaped distribution
(Gaussian)
occurring around a mean value. Special case of the binomial
distribution with n " 3 and p = 0.5.

f (x) = v 1 exp >− 1 d x −n 2 f(x) σ = 0.5
2r 2 v μ =0
nH m σ =1
v2 0 .5 σ =2
F (x) =

#x 1 exp >− 1 dt − n 2
2r 2 v
−3 v n Hdt x

–2 –1 0 1 2

Comment: Special case of the Normal (Gaussian) distribution. A unit normal table is included below.

Standardized f (x) = 1 exp d − x2 n n=0
(unit normal) 2r 2 v2 = 1
---
Hypergeometric #F(x) = x 1 t2
−3 2r exp d − 2 ndt
Poisson
Comment: Sample of dichotomous population (population of two types, e.g., defective/
not defective parts) without replacement. N is lot size, pN is number of defective parts

in the lot, P is the probability that in n random samples k will be defective.

d pN n= N (1 − p) G P(k)
k c n−k 0.4 p = 0.04
P (k) =
N p = 0.1
n m x=np 0.3

/F (x) d pN n= N (1 − p) G v2 = n p N − n (1 − p) p = 0.2
k n−k N − 1 0.2
= 0.1 N = 100
N
k # x c n m n = 20 k

0 5 10 15

Comment: P(k) is the probability that in n random samples k errors will occur. Used for
curves in a random sampling valuation. Conditions: large value of random samples
with a small value for proportion defective.

P (k) = (n p) k e−n $ p P(k)
k! 0.3 n.p = 1
x=np
/F (n p) k e−n $ p v2 = n p n.p = 5
k! 0.2 n.p =10
(x) =
0 .1
k # x

0 5 10 15 k

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Chapter 1: General Information

Probability and Density Functions (cont'd)

Kind of Probability Density Function f(x) Expected Mean (m), Form of the Density
Distribution Distribution Function F(x) Mean (x), Variance (s2) Function
Exponential
Comment: Special case of the Poisson distribution for x = 0 that gives the probability
Geometric without error. When used for reliability calculations, replace ^a $ xh with failure rate r
Negative
Binomial multiplied by control time t.

Gamma f (x) = a e−a x x = 1 2 f(x)
a20 a a =2
Weibull
x$0 v2 = 1 a =1
F (x) = 1 − e−a x a2 1 a =0.5
0 .5

0 0.5 1 2x

Comment: Describes the number of trials needed to get the first success, with p as
the success parameter.

f (x, p) = p (1 − p) x−1 n = 1
p
/x ---
v2 = 1 −p
F (x) = p (1 − p)n−1 p2

n=1

Comment: Describes the trial number of the kth success, with k as the stopping parameter
and p as the success probability.

P (k) = c n − 1 m pk (1 − p) n −k = 1
k − 1 p
x k

/F(x) = x n − 1 `1 − pj ---
k − 1 p2
c m pk (1 − p) n − k v2 = k

k<x

Comment: Gamma distribution is widely used to model physical quantities that take
positive values. The Gamma function is defined as

#C (k) = 3 xk−1 e−xdx where k $ 0, x $ 0
0

f (x) = bk 1 xk−1 e −x/b
C (k)
x = bk
Cc k, x m v2 = b2 k ---
b
F (x) = where k > 0
C (k) b>0

Comment: Note that when k = 1, the Weibull distribution reduces to the exponential
distribution with parameter 1.

f (t) = k t k − 1 e −c t k x = b Cc1 + 1 m
bk b k
m

F (t) = 1 − e −c t k v2 = b2 <Cc1 + 2 m − ---
b k
m

where 0 < t < 3 C2 c1 + 1 2
k
mG

©2020 NCEES 64

Chapter 1: General Information

Probability and Density Functions (cont'd)

Kind of Probability Density Function f(x) Expected Mean (m), Form of the Density
Distribution Distribution Function F(x) Mean (x), Variance (s2) Function

Triangular Comment: The triangular distribution is based on a simple geometric shape. The

Semicircle distribution arises naturally when uniformly distributed random variables are

U-Power transformed in various ways.
Distribution
f (x) = [\]]]]]]]]]]Z 2 2 (x − a) , a # x # a+p~
p~ 2 (a + ~ , a + p~ # x # a + w
− x)
2
p~ ~
3
]]\]]]]]]]][Z1p−1~~2 x = a + (1 + p)

F (x) = (x − a)2 , a # x # a+p~ v2 = ~2 [1 − p (1 − p)]
1 + a + p~ # x # a + 18
~ − 2
2 (1 − p) (a x) , w

Comment: The semicircular distribution is based on the shape of a semicircle with
center a (location parameter) and radius r (scale parameter).

f (x) = 2 r2 − (x − a)2
r r2

F (x) = 1 + x−a r2 − (x − a)2 + x=a
2 r r2
v2 = r2 ---
4
1 arcsin c x − a m
r r

where a − r # x # a + r

Comment: f(x) is symmetric about m.

f (x) = 2k + 1 d x − n 2k
2c c
n

= 1 >1 x − n 2k + 1 x=n ---
2 c
F (x) + d nH v2 = c2 2k + 1
2k + 3
where n − c # x # n − c

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Chapter 1: General Information

Normal Distribution Table
x f(x) F(x) R(x) 2 R(x) W(x)

0.0 0.3989 0.5000 0.5000 1.0000 0.0000
0.1 0.3970 0.5398 0.4602 0.9203 0.0797
0.2 0.3910 0.5793 0.4207 0.8415 0.1585
0.3 0.3814 0.6179 0.3821 0.7642 0.2358
0.4 0.3683 0.6554 0.3446 0.6892 0.3108
0.5 0.3521 0.6915 0.3085 0.6171 0.3829
0.6 0.3332 0.7257 0.2743 0.5485 0.4515
0.7 0.3123 0.7580 0.2420 0.4839 0.5161
0.8 0.2897 0.7881 0.2119 0.4237 0.5763
0.9 0.2661 0.8159 0.1841 0.3681 0.6319
1.0 0.2420 0.8413 0.1587 0.3173 0.6827
1.1 0.2179 0.8643 0.1357 0.2713 0.7287
1.2 0.1942 0.8849 0.1151 0.2301 0.7699
1.3 0.1714 0.9032 0.0968 0.1936 0.8064
1.4 0.1497 0.9192 0.0808 0.1615 0.8385
1.5 0.1295 0.9332 0.0668 0.1336 0.8664
1.6 0.1109 0.9452 0.0548 0.1096 0.8904
1.7 0.0940 0.9554 0.0446 0.0891 0.9109
1.8 0.0790 0.9641 0.0359 0.0719 0.9281
1.9 0.0656 0.9713 0.0287 0.0574 0.9426
2.0 0.0540 0.9772 0.0228 0.0455 0.9545
2.1 0.0440 0.9821 0.0179 0.0357 0.9643
2.2 0.0355 0.9861 0.0139 0.0278 0.9722
2.3 0.0283 0.9893 0.0107 0.0214 0.9786
2.4 0.0224 0.9918 0.0082 0.0164 0.9836
2.5 0.0175 0.9938 0.0062 0.0124 0.9876
2.6 0.0136 0.9953 0.0047 0.0093 0.9907
2.7 0.0104 0.9965 0.0035 0.0069 0.9931
2.8 0.0079 0.9974 0.0026 0.0051 0.9949
2.9 0.0060 0.9981 0.0019 0.0037 0.9963
3.0 0.0044 0.9987 0.0013 0.0027 0.9973

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Chapter 1: General Information

Normal Distribution Table (cont'd)

x f(x) F(x) R(x) 2 R(x) W(x)

1.2816 0.1755 Fractiles 0.8000
1.6449 0.1031 0.9000
1.9600 0.0584 0.9000 0.1000 0.2000 0.9500
2.0537 0.0484 0.9600
2.3263 0.0267 0.9500 0.0500 0.1000 0.9800
2.5758 0.0145 0.9900
0.9750 0.0250 0.0500

0.9800 0.0200 0.0400

0.9900 0.0100 0.0200

0.9950 0.0050 0.0100

1.3.4.7 Linear Regression and Goodness of Fit

Least Squares
y = at + btx

where y-intercept = at = yr - btxr
where
slope = bt = Sxy
Sxx

/ / /Sxy= n − c 1 mf n n yi p
=1 xi yi n =1
i xi pf
i
i−1

/ /Sxx= n xi2 − c 1 n 2
=1 n
mf xi p

i i=1

/yr= c 1 n yi p
n
mf

i=1

=/xr c 1 mf n xi p
n =1
i

n = sample size

Sxx = sum of squares of x
Syy = sum of squares of y
Sxy = sum of x-y products

Standard Error Estimate Se2

Se2 = SxxSyy − S 2 = MSE
xy

Sxx_n − 2i

where 2

/ /Syy= n yi2 − c 1 mf n yi p
=1 n =1
i i

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Chapter 1: General Information

Confidence Interval for Intercept at

at ! ta/2,n − 2 e 1 + xr2 oMSE
n Sxx

Confidence Interval for Slope bt

bt ! ta/2,n - 2 MSE
Sxx

Sample Correlation Coefficient R and Coefficient of Determination R2

R= Sxy R2 = S 2
SxxSyy xy

SxxSyy

1.3.4.8 Test Statistics

The following definitions apply:

Zvar = X - n o tvar = X - no
v s

nn

where

Zvar = the standard normal Z score
tvar = the sample distribution test statistic
v = known standard deviation

no = population mean

X = hypothesized mean or sample mean

n = sample size

s = computed sample standard deviation

The Z score is applicable when the standard deviations are known. The test statistic is applicable when the standard
deviations are computed at time of sampling.

Za corresponds to the appropriate probability under the normal probability curve for a given Zvar.
ta, n-1 corresponds to the appropriate probability under the t distribution with n-1 degrees of freedom for a given tvar.

Values of Za/2

Confidence Za/2
Interval

80% 1.2816

90% 1.6449

95% 1.9600

96% 2.0537

98% 2.3263

99% 2.5758

©2020 NCEES 68

©2020 NCEES 1.4 Chemistry and Physical Properties

1.4.1 Periodic Table of the Elements Periodic Table of E

I II Atomic Numbe
Symbol
1 4
Be Atomic Weigh
H 9.0122
69 21 22 23 24 25 26 27 2
1.0079 12 Sc Ti V Cr Mn Fe Co N
Mg 44.956 47.88 50.941 51.996 54.938 55.847 58.933 58
3 24.305 42 43
Li 39 40 41 Mo Tc 44 45 4
6.941 20 Y Zr Nb 95.94 (98) Ru Rh P
Ca 88.906 91.224 92.906 74 75 101.07 102.91 106
11 40.078 W Re
Na 57* 72 73 183.85 186.21 76 77 7
22.990 38 La Hf Ta Os Ir P
Sr 138.91 178.49 180.95 60 61 190.2 192.22 195
19 87.62 104 105 Nd Pm
K 89** Rf Ha 144.24 (145)
39.098 56 Ac (261) (262) 92 93
Ba 227.03 U Np
37 137.33 58 59 238.03 237.05
Rb Ce Pr
85.468 88 140.12 140.91
Ra 90 91
55 226.02 Th Pa
Cs 232.04 231.04
132.91

87
Fr
(223)

*Lanthanide Series 62 63 6
Sm Eu G
**Actinide Series 150.36 151.96 157

94 95 9
Pu Am C
(244) (243) (2

Elements III IV V VI VII VIII

er 5 6 7 8 9 2
ht B C N O F He
10.811 12.011 14.007 15.999 18.998 4.0026
28 29 30 Chapter 1: General Information
Ni Cu Zn 13 14 15 16 17 10
8.69 63.546 65.39 Al Si P S Cl Ne
26.981 28.086 30.974 32.066 35.453 20.179
46 47 48
Pd Ag Cd 31 32 33 34 35 18
6.42 107.87 112.41 Ga Ge As Se Br Ar
69.723 72.61 74.921 78.96 79.904 39.948
78 79 80
Pt Au Hg 49 50 51 52 53 36
5.08 196.97 200.59 In Sn Sb Te I Kr
114.82 118.71 121.75 127.60 126.90 83.80

81 82 83 84 85 54
Tl Pb Bi Po At Xe
204.38 207.2 208.98 (209) (210) 131.29

86
Rn
(222)

64 65 66 67 68 69 70 71
Gd Tb Dy Ho Er Tm Yb Lu
7.25 158.92 162.50 164.93 167.26 168.93 173.04 174.97

96 97 98 99 100 101 102 103
Cm Bk Cf Es Fm Md No Lr
247) (247) (251) (252) (257) (258) (259) (260)

Chapter 1: General Information

1.4.1.1 Relative Atomic Mass

Table of Relative Atomic Mass (Atomic Weight)

Name Symbol Atomic Atomic Name Symbol Atomic Atomic
Actinium Ac Number Mass Number Mass
Aluminum Al 164.930
Americium Am 89 ---* Holmium Ho 67 1.00797
Antimony Sb 1 114.82
Argon Ar 13 26.9815 Hydrogen H 49 126.9044
Arsenic As 53 192.2
Astatine At 95 ---* Indium In 77 55.847
Barium Ba 26 83.80
Berkelium Bk 51 121.75 Iodine I 36 138.91
Beryllium Be 57 207.19
Bismuth Bi 18 39.948 Iridium Ir 82 6.939
Boron B 3 174.97
Bromine Br 33 74.9216 Iron Fe 71 24.312
Cadmium Cd 12 54.9380
Calcium Ca 85 ---* Krypton Kr 25 ---*
Californium Cf 101 200.59
Carbon C 56 137.34 Lanthanum La 80 95.94
Cerium Ce 42 144.24
Cesium Cs 97 ---* Lead Pb 60 20.183
Chlorine Cl 10 ---*
Chromium Cr 4 9.0122 Lithium Li 93 58.71
Cobalt Co 28 92.906
Copper Cu 83 208.980 Lutetium Lu 41 14.0067
Curium Cm 7 ---*
Dysprosium Dy 5 10.811 Magnesium Mg 102 190.2
Einsteinium Es 76 15.9994
Erbium Er 35 79.904 Manganese Mn 8 106.4
Europium Eu 46 30.9738
Fermium Fm 48 112.40 Mendelevium Md 15 195.09
Fluorine F 78 ---*
Francium Fr 20 40.08 Mercury Hg 94 ---*
Gadolinium Gd 84 39.102
Gallium Ga 98 ---* Molybdenum Mo 19 140.907
Germanium Ge 59 ---*
Gold Au 6 12.01115 Neodymium Nd 61 ---*
Hafnium Hf 91 ---*
Helium He 58 140.12 Neon Ne 88 ---*
Rhodium Rh 86 186.2
Rubidium Rb 55 132.905 Neptunium Np 75 158.924
65 204.37
17 35.453 Nickel Ni 81

24 51.996 Niobium Nb

27 58.9332 Nitrogen N

29 63.546 Nobelium No

96 ---* Osmium Os

66 162.50 Oxygen O

99 ---* Palladium Pd

68 167.26 Phosphorus P

63 151.96 Platinum Pt

100 ---* Plutonium Pu

9 18.9984 Polonium Po

87 ---* Potassium K

64 157.25 Praseodymium Pr

31 69.72 Promethium Pm

32 72.59 Protactinium Pa

79 196.967 Radium Ra

72 178.49 Radon Rn

2 4.0026 Rhenium Re

45 102.905 Terbium Tb

37 85.47 Thallium Tl

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Chapter 1: General Information

Table of Relative Atomic Mass (Atomic Weight) (cont'd)

Name Symbol Atomic Atomic Name Symbol Atomic Atomic
Ruthenium Ru Number Mass Number Mass
Samarium Sm 232.038
Scandium Sc 44 101.07 Thorium Th 90 168.934
Selenium Se 69 118.69
Silicon Si 62 150.35 Thulium Tm 50 47.90
Silver Ag 22 183.85
Sodium Na 21 44.956 Tin Sn 74 238.03
Strontium Sr 92 50.942
Sulfur S 34 78.96 Titanium Ti 23 131.30
Tantalum Ta 54 173.04
Technetium Tc 14 28.086 Tungsten W 70 88.905
Tellurium Te 39 65.37
47 107.868 Uranium U 30 91.22
40
11 22.9898 Vanadium V

38 87.62 Xenon Xe

16 32.064 Ytterbium Yb

73 180.948 Yttrium Y

43 ---* Zinc Zn

52 127.60 Zirconium Zr

* Multiple isotopes

©2020 NCEES 71

Chapter 1: General Information

1.4.1.2 Oxidation Number Oxidation Number or Charge Number

Name Symbol Charge Name Symbol Charge
Acetate Fe +2, +3
Aluminum C2H3O2 –1 Iron Pb +2, +4
Ammonium Al +3 Lead Li
Barium Mg +1
Borate NH4 +1 Lithium Hg +2
Boron Ba +2 Magnesium Ni +1, +2
Bromine NO3 +2, +3
Calcium BO3 –3 Mercury NO2 –1
Carbon B +3 Nickel N –1
–3, +1, +2,
Carbonate Br –1 Nitrate O +3, +4, +5
Chlorate ClO4 –2
Chlorine Ca +2 Nitrite MnO4 –1
Chlorite PO4 –1
Chromate C +4, –4 Nitrogen –3
Chromium P –3, +3, +5
Copper CO3 –2 Oxygen K +1
Cyanide ClO3 –1 Perchlorate Si +4, –4
Dichromate –1 Permanganate Ag +1
Fluorine Cl –1 Phosphate Na +1
Gold –2 Phosphorus SO4 –2
Hydrogen ClO2 +2, +3, +6 Potassium SO3 –2
Hydroxide CrO4 +1, +2 Silicon S –2, +4, +6
Hypochlorite –1 Silver Sn +2, +4
Cr –2 Sodium Zn +2
–1 Sulfate
Cu +1, +3 Sulfite
+1 Sulfur
CN –1 Tin
–1 Zinc
Cr2O7
F

Au

H

OH

ClO

©2020 NCEES 72

Chapter 1: General Information

1.4.1.3 Organic Compounds

Families of Organic Compounds

FAMILY Specific IUPAC Name Common General Functional Group
Alkane Example Name Formula C–H and C–C
bonds
CH3CH3 Ethane Ethane RH
C =C
Alkene H2C = CH2 Ethene or Ethylene RCH = CH2
ethylene RCH = CHR – C=C–
R2C = CHR
R2C = CR2

Alkyne HC = CH Ethyne or Acetylene RC = CH
acetylene RC = CR

Arene Benzene Benzene ArH Aromatic ring

Haloalkane CH3CH2Cl Chloroethane Ethyl chloride RX CX

Alcohol CH3CH2OH Ethanol Ethyl alcohol ROH C OH

Ether CH3OCH3 Methoxymeth- Dimethyl ether ROR C OC
ane
Amine RNH2 CN
CH3NH2 Methanamine Methylamine R2NH
Aldehyde R3N O
O == = Ethanal Acetaldehyde = == == == CH
Ketone CH3CH O O
Carboxylic Acetone Dimethyl RCH C
Acid O ketone O
Ester CH3CCH3 O C OH
Ethanoic acid Acetic acid R1CR2 O
O C OC
CH3COH O
RCOH
O= Methyl Methyl =
CH3COCH3 ethanoate acetate O
RCOR

©2020 NCEES 73

Chapter 1: General Information

1.4.2 Industrial Chemicals Common Names of Industrial Chemicals

Common Name Chemical Name Molecular Formula
Acetone
Acetylene Acetone (CH3)2CO
Ammonia Acetylene C2H2
Ammonium Ammonia NH3
Anatase/rutile Ammonium hydroxide
Aniline Titanium dioxide NH4OH
Baking soda Aminobenzene TiO2
Battery acid Sodium bicarbonate
Sulfuric acid C6H5NH2
Bauxite NaHCO3
H2SO4
Bleach
Bleach Aluminum oxide Al2O3
Borane Hydrated aluminum oxide Al2O3 : 2H2O
Borax
Brine, salt Calcium hypochloride Ca(ClO)2
Carbide Sodium hypochlorite NaClO
Carbolic acid
Carbon dioxide Borane BH3
Carborundum
Caustic soda/lye Sodium tetraborate Na2B4O7 : 10H2O
Chalk Sodium chloride (solution)
Chlorite NaCl
Chlorate Calcium carbide
Cinnabar Phenol CaC2
Cumene C6H5OH
Deuterium Carbon dioxide
Dichromate Silicon carbide CO2
Dolomite Sodium hydroxide SiC
Epsom salt Calcium carbonate
Ether NaOH
Chlorite ion
Ethylene oxide CaCO3
Eyewash Chlorate ion ClO2–1
Formic acid ClO3–1
Glauber's salt Mercuric sulfide
Glycerine HgS
Grain alcohol
Graphite Isopropyl benzene C6H5CH(CH3)2
Gypsum Deuterium 2H

Dichromate ion Cr2O7-2
Magnesium carbonate MgCO3
MgSO4
Magnesium sulfate
(C2H3)2O
Diethyl ether

Ethylene oxide C2H4O
Boric acid (solution)
H3BO3
Methanoic acid HCOOH

Decahydrated sodium sulfate Na2SO4 : 10H2O
Glycerine
Ethanol C3H5(OH)3
C2H5OH
Crystalline carbon C

Calcium sulfate CaSO4 : 2H2O

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Chapter 1: General Information

Common Names of Industrial Chemicals (cont'd)

Common Name Chemical Name Molecular Formula

Heavy water Deuterium oxide (2H)2O
Hydronium Hydronium ion H3O+1
Hydroquinone P-dihydroxybenzene
Hypochlorite Hypochlorite ion C6H4(OH)2
OCl–1

Iron chloride Ferrous chloride FeCl2 : 4H2 O

Laughing gas Nitrous oxide N2O
Limestone Calcium carbonate CaCO3
Magnesia Magnesium oxide MgO

Magnetite Ferrous/ferric oxide Fe3O4
Marsh gas Methane CH4
Muriate of potash KCl
Potassium chloride

Muriatic acid Hydrochloric acid HCl

Neopentane 2,2-dimethylpropane CH3C(CH3)2CH3
Niter Sodium nitrate NaNO3
Niter cake Sodium bisulfate NaHSO4
Oleum
Ozone Fuming sulfuric acid SO3 in H2SO4
Perchlorate Ozone O3
Permanganate
Phosgene Perchlorate ion ClO4–1
Potash Permanganate ion MnO4–1
Potash
Phosgene COCl2
Potassium carbonate
Potassium hydroxide K2CO3
KOH

Prussic acid Hydrogen cyanide HCN

Pyrite, Fool's Gold Ferrous sulfide FeS

Pyrolusite Manganese dioxide MnO2
Quicklime Calcium oxide CaO

Quicksilver Mercury Hg

Sal soda/washing soda Decahydrated sodium carbonate Na2 CO3 : 10H2 O

Salammoniac Ammonium chloride NH4Cl
Salt/halite Sodium chloride NaCl

Salt cake Sodium sulfate (crude) Na2SO4
Sand/silica Silicon dioxide SiO2
Silane Silane SiH4
Slaked lime
Soda ash Calcium hydroxide Ca(OH)2
Styrene Sodium carbonate Na2CO3
Sugar C6H5CH=CH2
Vinyl benzene
Sucrose C12H22O11

Stannous chloride Stannous chloride SnCl2 : 2H2 O

Superphosphate Monohydrated primary calcium Ca^H2 PO4h2 : H2 O
phosphate

Toluene Methyl benzene C6H5CH3

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Chapter 1: General Information

Common Names of Industrial Chemicals (cont'd)

Common Name Chemical Name Molecular Formula

Trilene Tricholormethylene C2HCl3
Tritium Tritium 3H

Urea Urea (NH2)2CO
Vinegar (acetic acid) Ethanoic acid CH2COOH
Vinyl alcohol Vinyl alcohol CH2=CHOH
Vinyl chloride Vinyl chloride CH2=CHCl
Wood alcohol
Wolfram Methanol CH3OH
Tungsten W

Xylene Dimethyl benzene C6H4(CH3)2
Zinc blende Zinc sulfide ZiS

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2 MASS AND ENERGY BALANCES

2.1 Symbols and Definitions

Symbols

Symbol Description Units (U.S.) Units (SI)
H
Enthalpy Btu J

h Specific enthalpy Btu J = m2
lbm kg s2
lbm
MW Molecular weight (molar mass) lb mole kg
m Mass kg : mol
n Number of moles
P Pressure lbm kg

lb mole kg • mol

lbf or psi =Pa m=N2 kg
in 2 m : s2

Psat Saturation pressure, or vapor pressure lbf =Pa m=N2 kg
in 2 m : s2

p Partial pressure lbf =Pa m=N2 kg
Q Heat in 2 m : s2
S Entropy Btu
SG Specific gravity J
s Specific entropy†
Btu J
T Temperature oR K
t Time
U Internal energy dimensionless

Btu J
lbm-o R kg : K

°R or °F K or °C

hr s

Btu J

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Chapter 2: Mass and Energy Balances

Symbols (cont'd)

Symbol Description Units (U.S.) Units (SI)
J
u Specific internal energy† Btu kg
Volume lbm m3
V ft3 m3
kg
v Specific volume† ft 3
lbm
w
x Weight fraction dimensionless
x
y Mole fraction dimensionless

γB Quality dimensionless

ϕ Mole fraction dimensionless

r Mass concentration lbm kg
Volume fraction ft3 m3

dimensionless

Density lbm kg
ft3 m3

† Property values on molar basis are denoted by ^. For example, molar volume is vt.

2.2 Composition and Density

2.2.1 Measures of Composition

2.2.1.1 Mole, Mass, and Volume Fractions

Mole Fraction (or mole%): xi

xA = nA /n = i ni /i xi = 1
n
xA + xB = 1
For binary systems:

xA = na = 1 nB nA = c 1 − 1m nB
nA + nB na xB
1 +

Mass Fractions (Weight Fraction or wt%): wi

wA = mA /m = i mi /i wi = 1
m
wA + wB = 1
For binary systems:

wA = mA = 1 mB mA = c 1 − 1m mB
mA + mB mA wB
1 +

Conversion Between Mole Fraction and Mass Fraction

MWA = mA mA = nA MWA nA = mA
nA MWA

=i mmi AMMWWAi i wwi AMM=WWAi wA =i Nni AMMWWAi xA
/ / / /=xA MWi
i xi MWA

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Chapter 2: Mass and Energy Balances

For binary systems:

xA = mA MWA = 1 wA = nA MWB = 1
MWB MWA
mA + mB 1 + c 1 − 1m MWA nA + nB 1 + c 1 − 1 m MWB
wA MWB xA MWA

Volume Fraction (%vol): {i

{i = V * /where V[ = iVi[ /i {i = 1
i

V*

Volume fraction is the volume of a constituent of a mixture prior to mixing `Vi[j divided by the sum of volumes of all constitu-
ents prior to mixing `V [j.

For mixtures of ideal gases: φi = xi {i = wi t
For ideal solutions (no volume change due to mixing): ti

Density and Average Molecular Weight (MW) of a Mixture:

/MW = i xi MWi /1 = i wtii

For ideal solutions (no change in volume due to mixing): t

For solutions of components with similar densities (assume volume of the solution is proportional to the mass):

/t = i wi ti

Mass Fraction on a Dry Basis
For mixtures containing water, the mass fraction can be expressed on a dry basis, i.e., excluding the water.

Widry = `1 Wiwet
− WH2Oj

where WH2O is the mass fraction of water in the mixture.
2.2.1.2 Ratios or Loading

Mole Ratio: Xi

Ratios are used primarily for dilute solution or when one component is not affected by the process. For solutions with a solvent it
is also called "solute-free basis" and for combustion gases "dry basis."

Note: Component A is the basis (the solvent, the inert, or the predominant component).

=Xi nn=Ai xi /i ! A Xi = n − 1 XA = 1
xA nA

For binary systems (A: Solvent, B: Solute):

XB = 1 xB = 1 1 = 1 − 1 xB = 1 1 xA = 1 1
− xB xB xA XB + XB
− 1 1 +

For dilute systems with xA→1: Xi→xi

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Chapter 2: Mass and Energy Balances

Mass Ratio: Wi

=Wi mm=Ai wi /i ! AWi = m − 1 WA = 1
wA mA

For binary systems (A: Solvent, B: Solute):

WB = wB = 1 = 1 − 1 wB = 1 wA = 1
1 − wB 1 wA 1 1 + WB
wB − 1 1 + WB

For dilute systems with wA→1: Wi→wi

Conversion Between Mole Ratio and Mass Ratio

Wi = Xi MWi Xi = Wi MWA
MWA MWi

2.2.1.3 Concentrations

Molar Concentration: ci or [i]

ci = ni
V
p
For ideal gases: ci = xi RT

Mass Concentration: γi
=ci mV=i wi t

Volume Concentration: φi

zi = V [
i
V

Volume fraction is the volume of a constituent of a mixture prior to mixing V * divided by the volume of the mixture (V).
i

For mixtures in which volume decreases on mixing:

/i V * 2 Vmix /izi 2 1
i

Ideal solution (no volume change due to mixing):

z=i {=i t
wi ti

/i zi = 1 (ideal solutions only)

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Chapter 2: Mass and Energy Balances

2.2.1.4 Molarity and Molality

Molarity (M)

Molarity = gram : moles of solute
liters of solution

Note that molarity is temperature-dependent.

Molality (m)

Molality = gram : moles of solute
kg of solvent

Note that molality is temperature-independent.

2.2.1.5 Special Measures of Composition

Normality (N)

Normality = equivalent grams of the solute
liters of solution

Gram equivalent weight is a measure of the reactive capacity of a given molecule and thus is reaction-dependent.

Note that normality is temperature-dependent.

pH and pOH pH = − log10 8H3 O+B
pH = − log10 7H+A or
pOH =− log10 7OH−A

and

pK = pH + pOH =− log10K where K = 8H3 O+B 7OH−A

For water at 20cC: K = 10−14 and pK = 14

Note that all concentrations are in gram • moles/liter.

Proof (for Alcohol Content)

P=roof 2=abv 200 mL of pure ethanol
mL of solution

abv = alcohol % by volume (volume concentration)

For Dilute Solution (Can Be Based on Mass, Molar, or Volume)
ppm = parts per million = 10–6
ppb = parts per billion = 10–9
ppt = parts per trillion = 10–12
Percent: 1% = 10,000 ppm
Permil: 1a = 1000 ppm

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Chapter 2: Mass and Energy Balances

2.2.1.6 Conversion Table Between Different Measures of Concentration

Multicomponent Systems

Mole Mass Mole Ratio Mass Ratio Molar Mass
Fraction Xi Wi
Fraction concentration concentration
xi /Xi
wi ci gi
1 + j!A Xj
Mole wi MWi Wi ci MW
Fraction MW j /Xi MWi t MWi
xi /j w j /MWi+ j ! AWj MWi ci MW
xi = MWA + j ! A X j MWj MW j t
MWA
Mass
Fraction xi MW j /Wi ci Mt Wi cti
MWi ci
wi = /j x j wi 1 + j!AWj cA ci MWA
wi MWi c A MWi
Mole Ratio xi wA MWA Xi Wi MWA ci MWi
Xi = xA MWi cA MWA ci
wi cA
Mass Ratio xi MWA wA Xi MWi Wi
Wi = xA MWi wi t MWA
MWi
Molar xi t Xi cA Wi c A ci ci
concentration MW wi t MWi MWi

ci = xi MWi t Xi MWi cA wi cA ci MWi gi
MW
Mass
concentration / j!AWj

gi =

/Avg. MW j x j MWj 1 /MWA + j ! A X j MWj /1+ j!A Xj Wj
/1 +j!A Xj MW j
MW = / wj MWA

Avg.* MW j MWj
Density 1
/ x j MWj
r= / wj
j ti
j tj

*Ideal solutions only

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Chapter 2: Mass and Energy Balances

Binary Systems

Mole Mass Mole Ratio Mass Ratio Molar Mass
Fraction Fraction XB WB concentration concentra-

xB wB cB tion
gB
Mole wB MWB XB WB cB MW
Fraction MWA 1 + XB t cB MW
xB wB + _1 − wB i MWB + WB t MWB
xB = MWA

Mass xB MWA XB WB cB Mt WB ctB
Fraction MWB MWA 1 + WB
xB + _1 − xBi wB MWB + XB cB
wB = cA
cB MWB
Mole Ratio xB wB MWA XB WB MWA cA MWA cB MWA
XB = 1 − xB 1 − wB MWB MWB c A MWB
cB
Mass Ratio xB MWB wB XB MWB WB c B
WB = 1 − xB MWA 1 − wB MWA c A

Molar xB t wB t XB cA WB c A cB
concentration MW MWB MWB MWB
wB t
cB = xB MWB t XB cA MWB wB cA cB MWB gB
MW MWA + XB MWB
Mass
concentration 1 + XB

gB =

Avg. MW MWB MWB 1 + WB
MW = MWA
xB MWB + _1 − xBi MWA wB + _1 − w Bi 1 + WB
MWA MWB

Avg.* tB MW tB tB d MW + XBn _1 + WBitB
Density MWB _1 − w MWB
MWA t t B tB
r= xB + _1 − xBi MWB t B wB + Bi t A XB + MWA t B WB + tA
A MWB t A

*Ideal solutions only
Note: For mole and mass ratios, "A" is the basis component (e.g., the solvent).

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Chapter 2: Mass and Energy Balances

2.2.2 Density

2.2.2.1 Density and Relative Density

Density is

t = m
V

Relative density is

RD = t
tref

where tref = density of a reference material

2.2.2.2 Specific Gravity

Specific Gravity (Relative Density) of Gas

SG = tair at tgas

ref temp,press

The reference temperatures are commonly either 0°C or 60°F and the reference pressure is commonly 14.696 psia (101,325 Pa).

For ideal gas:

SG = MWgas = MWgas
MWair
28.96 g
mol

Specific Gravity (Relative Density) of Liquid

SG = t temp
tH2O at ref

where t=H 2 O, 4cC 6=2.4 lfbtm3 1000 kg
m3

The reference temperatures are commonly either 4°C or 60°F.

Specific Gravity (Relative Density) in Baumé

For liquids lighter than water, using degrees Baumé or B°: SG = 140
130 + Bc

For liquids heavier than water, using degrees Baumé or B°: SG = 145
145 − Bc

Specific Gravity (Relative Density) for Hydrocarbon Liquid

SG60cF = 141.5 API = 141.5 − 131.5
131.5 + API SG60cF

where API = American Petroleum Institute gravity or API gravity

Specific Gravity (Relative Density) for Slurries

Bulk density and specific gravity of solids and liquid mixtures (slurries) are

1 = 1 + \solidsd 1 − 1 n
tbulk tliquid tsolids tliquid

1 = 1 + \solidse 1 − 1 o
SGbulk SGliquid SGsolids SGliquid

where xsolids is the mass fraction of the solid in the slurry.

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Chapter 2: Mass and Energy Balances

2.3 Mass Balance

General balance equation: Accumulation = Input – Output + Generation – Consumption

2.3.1 Mass Balances Without Reaction

Balanced equation at steady state with no reaction: Input = Output

2.3.1.1 Purge, Bypass and Recycle Configurations

Common Flow Configurations

FEED PRODUCT FEED PRODUCT

BYPASS RECYCLE
PROCESS WITH BYPASS PROCESS WITH RECYCLE

FEED PRODUCT FEED PRODUCT
PURGE PURGE
RECYCLE
PROCESS WITH PURGE PROCESS WITH RECYCLE

AND PURGE

2.3.2 Mass Balances With Reaction

Balanced equation at steady state with reaction:

Input + Generation = Output + Consumption

2.3.2.1 Stoichiometry
The stoichiometric amount of each reactant is the exact amount needed for the reaction to go to 100% conversion with no
reactants left over. Such a reaction makes the stoichiometric amount of each product.

2.3.2.2 Combustion Reactions
Theoretical (stoichiometric) air is the minimum theoretical air required for complete combustion.

Molar air-fuel ratio c A m = Moles of air
F Moles of fuel

c A m
F
Percent theoretical air = Actual # 100

c A m
F
Theoretical

c A m − c A m
F F
= Actual Theoretical # 100

Percent excess air c A m
F
Theoretical

Gross or higher heating value (HHV) is the heat of combustion assuming all water generated is condensed as a liquid.
Net or lower heating value (LHV) is the heat of combustion assuming that no water is condensed.

©2020 NCEES 85

Chapter 2: Mass and Energy Balances

Major Components of Air

Element Volume, %
Nitrogen 78.09
Oxygen 20.94
0.93
Argon

The dry adiabatic lapse rate ΓAD is 0.98°C per 100 m (5.4°F per 1000 ft). This is the rate at which dry air cools adiabatically with
altitude. The actual (environmental) lapse rate Γ is compared to ΓAD to determine stability.

Stability of Adiabatic Lapse Rate

Lapse Rate Stability Condition

G > GAD Unstable
G = GAD Neutral
G < GAD Stable

2.4 Energy Balances

2.4.1 Energy Balances without Reaction

2.4.1.1 Sensible Heat

Qo = m cp dT
dt

cp = DH
m DT

Heat transferred in or out of a flowing material:

Qo = mo cp DT
Note: The dot superscript (e.g., Qo and mo ) indicates a rate.

2.4.1.2 Heat of Solution

Ideal mixing applies to gases at low pressures; liquids and high-pressure gases involve nonideal mixing. In these cases, make
calculations on a mole or mass basis instead of on a mole-fraction or mass-fraction basis. For the heat of a solution for a binary
mixture on a molar basis:

n htmix, actual = n Dht + n1 ht1 + n2 ht2

where

n = total moles of solution

n1 = moles of Component 1

n2 = moles of Component 2
This equation also applies to solids or gases dissolving into liquids. The Dht value must be known.

Heats of solutions often appear in charts, and enthalpies of mixing are presented as a function of composition. For evolved or
absorbed heat:

n Dht = n htmix, final - _n1 htmix1 + n2 htmix2i

where hmix1 and hmix2 can be either mixtures or pure components. This is calculated on a mass basis if the data are on a mass basis.

©2020 NCEES 86

Chapter 2: Mass and Energy Balances

2.4.1.3 Vapor-Liquid Systems

Quality x (for liquid-vapor systems at saturation) is defined as the mass fraction of the vapor phase:

x = mv
mv + ml

where

mv = mass of vapor
ml = mass of liquid
Note: Quality for steam might be expressed as a percentage. Moisture is the fraction of mass in a liquid phase.

Moisture = 1 – x

Specific volume of a two-phase system can be represented as

v = xvv + (1 – x) vl or v = v1 + xDvvap
where

vv = specific volume of saturated vapor

vl = specific volume of saturated liquid

Dvvap = specific volume change upon vaporization

= vv – vl

uSimilar expressions exist for , h, and s:
u u u u u u= x v + (1 – x) l or
= l + xD vap

h = xhv + (1 – x) hl or h = hl + xDhvap

s = xsv + (1 – x) sl or s = sl + xDsvap

The energy difference between two phases in equilibrium at a given temperature (or pressure) is the latent heat. The three types of
latent heat are

Latent heat of fusion (melting): Dhfusion = hl – hs
Latent heat of sublimation: Dhsubl = hv – hs
Latent heat of vaporization: Dhvap = hv – hl

2.4.2 Energy Balances with Reaction

2.4.2.1 Heat of Reaction

Calculate standard state heat of reaction Dht 0 from standard heat of formation Dht 0 at 298 K (25°C) and 1 atm, using
R f

/ /Dht0= Dht 0 − Dht 0
R f f
products reactants

Calculate DhtR at temperature T using
/ /DhtR
= Dht 0 + Dhtf + Dhtf
R

products reactants

where Dhtf includes the sensible and latent heat changes between T and 298K

©2020 NCEES 87

Chapter 2: Mass and Energy Balances

2.4.2.2 Heat of Formation and Heat of Combustion
The standard heats of formation and combustion at 25°C are shown in the tables below. The products of combustion are H2O (l)
and CO2 (g). Solids are listed as s in the tables below.

Heats of Formation and Heats of Combustion for Alkanes

Dht 0 − Dht 0
f c

Name Formula Phase HHV

Methane CH4 g kJ Btu kJ Btu
Ethane C2H6 g mol lb mol mol lb mol
n-Propane C3H8 g –74.6 –32,070 890.7 382,900
Isobutane C4H10 g –84.00 –36,110
n-Butane C4H10 g –104.6 –44,970 1560 670,700
n-Pentane C5H12 g –134.3 –57,740
n-Pentane C5H12 l –125.5 –53,960 2219 954,100
Cyclohexane C6H12 g –146.9 –63,160
Cyclohexane C6H12 l –173.5 –74,600 2868 1,233,000
n-Hexane C6H14 g –124.0 –53,310
n-Hexane C6H14 l –157.0 –67,500 2877 1,237,000
Methylcyclohexane C7H14 g –167.2 –71,890
Methylcyclohexane C7H14 l –198.8 –85,470 3535 1,520,000
n-Heptane C7H16 g –154.78 –66,540
n-Heptane C7H16 l –190.2 –81,760 3509 1,507,000
n-Octane C8H18 g –187.9 –80,790
n-Octane C8H18 l –225.0 –96,740 ——
n-Nonane C9H20 g –208.4 –89,600
n-Nonane C9H20 l –250.0 –107,500 3930 1,690,000
n-Decane C10H22 g –228.3 –98,160
n-Decane C10H22 l –274.7 –118,100 4199 1,805,000
–249.7 –107,400
–301.0 –129,400 4163 1,790,000

4601 1,978,000

4565 1,963,000

——

4817 2,071,000

——

5430 2,335,000

——

6125 2,633,000

——

6779 2,915,000

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Chapter 2: Mass and Energy Balances

Heats of Formation and Heats of Combustion for Alkenes and Alkynes

Dht 0 − Dht 0
f c

Name Formula Phase HHV

Acetylene kJ Btu kJ Btu
Ethylene mol lb mol mol lb mol
Propylene 226.8 97,510
1,3-Butadiene C2H2 g 52.3 22,500 ——
1,3-Butadiene C2H4 g 20.4
1-Butene C3H6 g 109 8770 1411 606,600
1-Pentene C4H6 g 91 46,900
1-Pentene C4H6 l –0.630 39,100 2058 884,800
1-Hexene C4H8 g –22
1-Hexene C5H10 g –49 –270 2540 1,092,000
C5H10 l –42 –9500
C6H12 g –73 –21,000 2522 1,084,000
C6H12 l –18,000
–31,000 2717 1,168,000

——

3350 1,440,000

——

——

Heats of Formation and Heats of Combustion for Aromatics

Dht 0 − Dht 0
f c

Name Formula Phase HHV

Benzene C6H6 g kJ Btu kJ Btu
Benzene C6H6 l mol lb mol mol lb mol
Toluene C7H8 g 82.9 35,600 ——
Toluene C7H8 l 49 21,000
Styrene C8H8 g 50 21,000 3270 1,406,000
Styrene C8H8 l 12
Ethylbenzene C8H10 g 147 5200 ——
Ethylbenzene C8H10 l 103 63,200
p-Xylene C8H12 g 49 44,300 3920 1,685,000
p-Xylene C8H12 l 21,000
o-Xylene 6.8 ——
o-Xylene C8H12 g 17.9 2900
C8H12 l –24.4 7700 4390 1,887,000
–10,500
19 ——
–24.4 8200
–10,500 4567 1,964,000

——

4552 1,957,000

——

4552 1,957,000

©2020 NCEES 89