j i
⫺
v v
v v
⭈⭈⭈
⭈⭈⭈
⫺
Figure 1.14
d d d d (1 40)
Dividing equation 1.38 by the total number of moles in the solution yields (1 41) where is the molar volume of the solution and is the mole fraction of sub- stance in the solution. In Chapter 6 we will discuss the determination of the partial molar volume of a species in a solution, and we will also see that in ideal solutions the partial molar volume of a substance is equal to its molar volume in the pure liquid.
In applying thermodynamics we generally ignore the effect of the gravitational field, but it is important to realize that if there is a difference in height there is a difference in gravitational potential. For example, consider a vertical column of a gas with a unit cross section and a uniform temperature , as shown in Fig. 1.14.
The pressure at any height is simply equal to the mass of gas above that height per unit area times the gravitational acceleration . The standard acceleration due to gravity is defined as 9 806 65 m s . The difference in pressure d between and d is equal to the mass of the gas between these two levels times and divided by the area. Thus,
d d (1 42)
Pressure and composition of air at 10 km
Comment:
PM RT M
P PMg h .
RT
h P h
P
P gM
h .
P RT
P gMh
P RT .
P P .
This is our first contact with exponential functions, but there will be many more.
The barometric formula can also be regarded as an example of a Boltzmann distribution, which will be deri ed in Chapter 16 (Statistical Mechanics). The temperature determines the way particles distribute themsel es o er arious energy le els in a system.
P h
P
gMh RT
25
P P gMh
RT
P . .
. .
P . .
. .
y . y .
0
2
2
2 2
0
0
0 0 /
2 2
0 2
2 3 1 4
O 1 1
2
3 4
N
O N
Assuming that air is 20% O and 80% N at sea level and that the pressure is 1 bar, what are the composition and pressure at a height of 10 km, if the atmosphere has a temperature of 0 C independent of altitude?
exp For O ,
(9 8 m s )(32 10 kg mol )(10 m) (0 20 bar)exp
(8 3145 J K mol )(273 K) 0 0503 bar
For N ,
9 8 28 10 10
(0 80)exp
8 3145 273 0 239 bar
The total pressure is 0.289 bar, and 0 173 and 0 827.
⫺
⫺ ⫻
⫻ ⫻ ⫻
⫺ ⫻
barometric formula.
Example 1.11
1.11 Special Topic: Barometric Formula
冢 冣
冤 冥
冢 冣
冮 冮
⫺
⫺
⫺ ⫺
⫺ ⫺
⫺
⬚
v
v v v v
⫺
⫺
⫺
where is the density of the gas. If the gas is an ideal gas, then / , where is the molar mass, so that
d d (1 43)
Separating variables and integrating from 0, where the pressure is , to , where the pressure is , yields
d d (1 44)
ln (1 45)
e (1 46)
This relation is known as the
Figure 1.15 gives the partial pressures of oxygen, nitrogen, and the total pres- sure as a function of height in feet, assuming the temperature is 273.15 K inde- pendent of height.
1 0.8 0.6 0.4 0.2
10 000 Ptotal PN2
PO2
20 000 30 000 40 000 50 000 h/feet
P/bar
N N
Z
F N
F N
F N D
F p p
Nine Key Ideas in Chapter 1
s s
s s s
Partial pressures of oxygen, nitrogen, and the total pressure of the atmo- sphere as a function of height in feet, assuming the temperature is 273.15 K independent of height (see Computer Problem 1.H).
1.
2.
3.
4.
5.
6.
7.
8.
9.
⫺ Figure 1.15
The state of a macroscopic system at equilibrium can be specified by the values of a small number of macroscopic variables. For a system in which there are no chemical reactions, the intensive state of a one-phase system can be specified by 1 intensive variables, where is the number of different species.
According to the zeroth law of thermodynamics, if systems A and B are individually in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other.
The ideal gas temperature scale is based on the behavior of gases in the limit of low pressures. The unit of thermodynamic temperature, the kelvin, represented by K, is defined as the fraction 1/273.16 of the temperature of the triple point of water.
The total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the gases in the mixture.
The virial equation of state, which expresses the compressibility factor for a gas in terms of powers of the reciprocal molar volume or of the pres- sure, is useful for expressing experimental data on a gas provided the pres- sure is not too high or the gas too close to its critical point.
The van der Waals equation is useful because it exhibits phase separation between gas and liquid phases, but it does not represent experimental data exactly.
For a one-phase system without chemical reactions, we have seen that the number of degrees of freedom is equal to 1. But if the system con- tains two phases at equilibrium, , and if the system contains three phases at equilibrium, 1. The number of variables required to describe the extensive state of a multiphase macroscopic system at equi- librium is , where is the number of phases.
The volume of a mixture is equal to the sum of the partial molar volumes of the species it contains each multiplied by the amount of that species.
For an isothermal atmosphere, the pressure decreases exponentially with the height above the surface of the earth.
y y
y
i i
27
A Sur ey of Thermodynamics.
Thermodynamics for Chemical Engineers.
The Virial Coefficients of Pure Gases and Mixtures.
Thermodynamics,
Introduction to Chemical Engineering Thermodynamics.
Thermodynamics and Its Applications.
P P
T P T V P V P
.
i y P .
y P
a b t
c
A Bt
A B
V
a b b
b P
P n
Z P a b
Z P P n T . P .
3 3
1
3 l v
2 2
2 2
2 2 l
v
4 3
l v
2 4 2
3 c
2 4
2 4 2
2 4
c c
M. Bailyn, New York: American Institute of Physics, 1944.
K. E. Bett, J. S. Rowlinson, and G. Saville, Cam-
bridge, MA: MIT Press, 1975.
J. H. Dymond and E. B. Smith, Oxford,
UK: Oxford University Press, 1980.
K. S. Pitzer, 3rd ed. New York: McGraw-Hill, 1995.
J. M. Smith, H. C. Van Ness, and M. M. Abbott, New York: McGraw-Hill, 1996.
J. W. Tester and M. Modell, Upper Saddle River,
NJ: Prentice PTR, 1997.
Show how the second virial coefficient of a gas and Problems marked with an icon may be more conve-
its molar mass can be obtained by plotting / versus , where niently solved on a personal computer with a mathematical pro-
is the density of the gas. Apply this method to the following gram.
data on ethane at 300 K.
The intensive state of an ideal gas can be completely de-
fined by specifying (1) , , (2) , , or (3) , . The extensive /bar 1 10 20
state of an ideal gas can be specified in four ways. What are the /10 g cm 1.2145 13.006 28.235 combinations of properties that can be used to specify the exten-
Calculate the second and third virial coefficients for sive state of an ideal gas? Although these choices are deduced
hydrogen at 0 C from the fact that the molar volumes at for an ideal gas, they also apply to real gases.
50.7, 101.3, 202.6, and 303.9 bar are 0.4634, 0.2386, 0.1271, and The ideal gas law also represents the behavior of mixtures 0 090 04 L mol , respectively.
of gases at low pressures. The molar volume of the mixture is
the volume divided by the amount of the mixture. The partial The critical temperature of carbon tetrachloride is pressure of gas in a mixture is defined as for an ideal gas 283 1 C. The densities in g/cm of the liquid and vapor mixture, where is its mole fraction and is the total pressure. at different temperatures are as follows:
Ten grams of N is mixed with 5 g of O and held at 25 C at 0.750
/ C 100 150 200 250 270 280
bar. ( ) What are the mole fractions of N and O ? ( ) What are
1.4343 1.3215 1.1888 0.9980 0.8666 0.7634 the partial pressures of N and O ? ( ) What is the volume of the
0.0103 0.0304 0.0742 0.1754 0.2710 0.3597 ideal mixture?
What is the critical molar volume of CCl ? It is found that the A mixture of methane and ethane is contained in a glass
mean of the densities of the liquid and vapor does not vary bulb of 500 cm capacity at 25 C. The pressure is 1.25 bar, and
rapidly with temperature and can be represented by the mass of gas in the bulb is 0.530 g. What is the average molar
mass, and what is the mole fraction of methane?
Nitrogen tetroxide is partially dissociated in the gas phase according to the reaction 2
where and are constants. The extrapolated value of the av- N O (g) 2NO (g)
erage density at the critical temperature is the critical density.
The molar volume at the critical point is equal to the molar A mass of 1.588 g of N O is placed in a 500-cm glass vessel at
mass divided by the critical density.
298 K and dissociates to an equilibrium mixture at 1.0133 bar.
( ) What are the mole fractions of N O and NO ? ( ) What Show that for a gas of rigid spherical molecules, in percentage of the N O has dissociated? Assume that the gases the van der Waals equation is four times the molecular vol-
are ideal. ume times Avogadro’s constant. If the molecular diameter of
Ne is 0.258 nm (Table 17.4), approximately what value of is Although a real gas obeys the ideal gas law in the limit
expected?
as 0, not all of the properties of a real gas approach the
values for an ideal gas as 0. The second virial coefficient What is the molar volume of -hexane at 660 K and 91 of an ideal gas is zero, and so d /d 0 at all pressures. But bar according to ( ) the ideal gas law and ( ) the van der Waals calculate d /d for a real gas as 0. equation? For -hexane, 507 7 K and 30 3 bar.
v
Problems
REFERENCES
PROBLEMS
⫺ ⫺
⫺
⬚
⬚
⬚ ⬚
⬚
1.6
1.1
1.7 1.2
1.8
1.3
1.4
1.9
1.5
1.10
N N i j ij
i j
P ij
ij ji
T
T V
a
b a
b . c
a b a c
. b .
Z P
V P T
N
B y y B
y i j
V
V T i j B
B B B
y y B B B
V
V P
t
T
Z P
a P V P T
P nRT
V nb
b P V T P T V a
b c
n
a b
a h b
2 2
2
1 1 2
6 6 3 2 2 2
1
3
2
1 1
1 2 11 12 22
3
l v
4 0
2 2
1
2 2
Derive the expressions for van der Waals constants and Calculate the pressure due to a mass of 100 kg in the in terms of the critical temperature and pressure; that is, derive earth’s gravitational field resting on an area of ( ) 100 cm and equations 1.32 and 1.33 from 1.29–1.31. ( ) 0 01 cm . ( ) What area is required to give a pressure of 1
Calculate the second virial coefficient of methane at 300 K bar?
and 400 K from its van der Waals constants, and compare these A mole of air (80% nitrogen and 20% oxygen by vol- results with Fig. 1.9. ume) at 298.15 K is brought into contact with liquid water, which has a vapor pressure of 3168 Pa at this temperature.
You want to calculate the molar volume of O at 298.15
( ) What is the volume of the dry air if the pressure is 1 bar?
K and 50 bar using the van der Waals equation, but you don’t
( ) What is the final volume of the air saturated with water va- want to solve a cubic equation. Use the first two terms of
por if the total pressure is maintained at 1 bar? ( ) What are the equation 1.26. The van der Waals constants of O are
mole fractions of N , O , and H O in the moist air? Assume the 0 138 Pa m mol and 31 8 10 m mol . What is the
gases are ideal.
molar volume in L mol ?
Using Fig. 1.9, calculate the compressibility factor for The isothermal compressibility of a gas is defined in
NH (g) at 400 K and 50 bar.
Problem 1.17, and its value for an ideal gas is shown to be 1/ .
Use implicit differentiation of with respect to at constant In this chapter we have considered only pure gases, but to obtain the expression for the isothermal compressibility of a it is important to make calculations on mixtures as well. This van der Waals gas. Show that in the limit of infinite volume, the requires information in addition to that for pure gases. Statis- value for an ideal gas is obtained. tical mechanics shows that the second virial coefficient for an
-component gaseous mixture is given by Calculate the second and third virial coefficients of O
from its van der Waals constants in Table 1.3.
Calculate the critical constants for ethane using the van der Waals constants in Table 1.3.
The cubic expansion coefficient is defined by
where is mole fraction and and identify components. Both 1 indices run over all components of the mixture. The bimolecular interactions between and are characterized by , and so
. Use this expression to derive the expression for and the isothermal compressibility is defined by for a binary mixture in terms of , , , , and .
1 The densities of liquid and vapor methyl ether in
g cm at various temperatures are as follows:
Calculate these quantities for an ideal gas. / C 30 50 70 100 120
What is the equation of state for a liquid for which the co- 0.6455 0.6116 0.5735 0.4950 0.4040 efficient of cubic expansion and the isothermal compressibility 0.0142 0.0241 0.0385 0.0810 0.1465
are constant?
The critical temperature of methyl ether is 299 C. What is the For a liquid the cubic expansion coefficient is nearly
critical molar volume? (See Problem 1.8.) constant over a narrow range of temperature. Derive the expres-
Use the van der Waals constants for CH in Table 1.3 to sion for the volume as a function of temperature and the limiting
calculate the initial slopes of the plots of the compressibility fac- form for temperatures close to .
tor versus at 300 and 600 K.
( ) Calculate ( / ) and ( / ) for a gas that has
A gas follows the van der Waals equation. Derive the rela- the following equation of state:
tion between the third and fourth virial coefficients and the van der Waals constants.
Using the van der Waals equation, calculate the pressure exerted by 1 mol of carbon dioxide at 0 C in a volume of ( ) 1.00 ( ) Show that ( / ) ( / ). These are referred
L and ( ) 0.05 L. ( ) Repeat the calculations at 100 C and 0.05 L.
to as mixed partial derivatives.
A mole of -hexane is confined in a volume of 0.500 L at Assuming that the atmosphere is isothermal at 0 C and
600 K. What will be the pressure according to ( ) the ideal gas that the average molar mass of air is 29 g mol , calculate the
law and ( ) the van der Waals equation? (See Problem 1.10.) atmospheric pressure at 20 000 ft above sea level.
A mole of ethane is contained in a 200-mL cylinder at 373 Calculate the pressure and composition of air on the top
K. What is the pressure according to ( ) the ideal gas law and of Mt. Everest, assuming that the atmosphere has a temperature
( ) the van der Waals equation? The van der Waals constants of 0 C independent of altitude ( 29 141 ft). Assume that air
are given in Table 1.3.
at sea level is 20% O and 80% N .
⫻
⭸
⭸
⫺ ⭸⭸
⭸ ⭸ ⭸ ⭸
⫺
⭸ ⭸ ⭸ ⭸ ⭸ ⭸
␣
␣
␣
␣
冱冱
冢 冣 冢 冣
⫺ ⫺ ⫺
⫺
⫺
⫺
⬚
⬚
⬚ ⬚
⬚
⬚
1.11 1.23
1.12
1.24 1.13
1.14 1.25
1.26
1.15 1.16 1.17
1.27
1.18
1.19
1.28 1.20
1.29
1.30
1.21 1.31
1.22 1.32
Computer Problems
T
T
B . . . .
29
a P V
V P b
c P
P V b RT a
b
a b c
B
a
b P
m V
M M RT P
a
B T T
b c
B . C .
3 1
⬘ ⫺
1
2 2 1
2 2
2 2
2
/K 75 100 125 150 200 250 300 400 500 600 700 /cm mol 274 160 104 71 5 35 2 16 2 4 2 9 16.9 21.3 24
2
2 4 3
2 4 2
1
1
4 2 2
2
When pressure is applied to a liquid, its volume decreases. ( ) Plot the pressure of ethane versus its molar volume in Assuming that the isothermal compressibility the range 0 200 bar and molar volumes up to 0.5 mol L
using the van der Waals equation at 265, 280, 310.671, 350, and 1 400 K, where 310.671 K is the critical temperature calculated with the van der Waals constants. ( ) Discuss the significance of the plots and the extent to which they represent reality. ( ) is independent of pressure, derive an expression for the volume
Calculate the molar volumes at 400 K and 150 bar and at as a function of pressure.
265 K and 20 bar.
Calculate and for a gas for which
This is a follow-up to Computer Problem 1.B on the van der Waals equation. ( ) Plot the derivative of the pressure with
( )
respect to the molar volume for ethane at 265 K. ( ) Plot the What is the molar volume of N (g) at 500 K and 600 bar derivative at the critical temperature. ( ) Plot the second deriva- according to ( ) the ideal gas law and ( ) the virial equation? tive of the pressure with respect to the molar volume at the criti- The virial coefficient of N (g) at 500 K is 0.0169 L mol . cal temperature. In each case, what is the significance of the
What is the mean atmospheric pressure in Denver, Col- maxima and minima?
orado, which is a mile high, assuming an isothermal atmosphere ( ) Express the compressibility factors for N and O at at 25 C? Air may be taken to be 20% O and 80% N . 298.15 K as a function of pressure using the virial coefficients in Calculate the pressure and composition of air 100 miles Table 1.1. ( ) Plot these compressibility factors versus from 0 above the surface of the earth assuming that the atmosphere has to 1000 bar.
a temperature of 0 C independent of altitude. The second virial coefficients of N at a series of tempera- The density / of a mixture of ideal gases A and tures are given by
B is determined and is used to calculate the average molar mass of the mixture; / . How is the average molar mass determined in this way related to the molar masses of A and B?
( ) Fit these data to the function Figure 1.13 shows the Maxwell construction for cal-
culating the vapor pressure of a liquid from its equation of state.
Since this requires an iterative process, a computer is needed,
( ) Plot this function versus temperature. ( ) Calculate the and J. H. Noggle and R. H. Wood have shown how to write
Boyle temperature of molecular nitrogen.
a computer program in Mathematica (Wolfram Research, Inc.,
Nitrogen tetroxide (N O ) gas is placed in a 500-cm glass Champaign, IL 61820-7237) to do this. Use this method with the
vessel, and the reaction N O 2NO goes to equilibrium at van der Waals equation to calculate the vapor pressure of nitro-
25 C. The density of the gas at equilibrium at 1.0133 bar is gen at 120 K.
3.176 g L . Assuming that the gas mixture is ideal, what are the partial pressures of the two gases at equilibrium?
Calculate the molar volume of ethane at 350 K and 70 bar using the van der Waals constants in Table 1.3.
Problem 1.7 yields 0 135 L mol and 4 3 Plot the partial pressures of oxygen, nitrogen, and the to- 10 L mol for H (g) at 0 C. Calculate the molar volumes of tal pressure in bars versus height above the surface of the earth molecular hydrogen at 75 and 150 bar and compare these molar from zero to 50 000 feet assuming that the temperature is con- volumes with the molar volume of an ideal gas. stant at 273 K.
⬍ ⬍
⫺ ⭸⭸
⫺
⫻
␣
␣  ␥
Problems
冢 冣
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⫺ ⫺ ⫺ ⫺ ⫺ ⫺ ⫺
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⬘
⬚
⬚
⬚
⬚
1.33 1.B
1.34 1.C
1.35
1.36
1.D 1.37
1.E 1.38
1.39
1.F
1.G
1.A 1.H
U
H U P V
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13
2
Work and Heat
First Law of Thermodynamics and Internal Energy
Exact and Inexact Differentials Work of Compression and Expansion of a Gas at Constant Temperature Various Kinds of Work
Change in State at Constant Volume Enthalpy and Change of State at Constant Pressure
Heat Capacities
Joule Thomson Expansion Adiabatic Processes with Gases Thermochemistry
Enthalpy of Formation Calorimetry
–
In this chapter we begin to emphasize processes that take a chemical system from one state to another. The first law of thermodynamics, which is often referred to as the law of conservation of energy, leads to the definition of a new thermodynamic state function, the internal energy . An additional state function, the enthalpy
, is defined in terms of , , and for reasons of convenience.
Thermochemistry, which deals with the heat produced by chemical reactions and solution processes, is based on the first law. If heat capacities of reactants and products are known, the heat of a reaction may be calculated at other tempera- tures once it is known at one temperature.
First Law of Thermodynamics
w is positive
System m
System m
(a)
w is negative System
m
System m
(b) h
Vacuum Vacuum
Vacuum Vacuum
h
f a
f a
f L
f L
m .
m w
w .
f L
fL
w f L
f L
P PA A
L PA L A L V
P V w
w
q q q
PV
w P V .
P
m
g mgh h
w w
31
†
a
b
*Since this is our first contact with vectors and matrices, we want to note that they are represented by boldface italic type. They may have units like other physical quantities. Sections D.7 and D.8 of Appendix D give information about the mathematical properties of vectors and matrices.
We use d rather than d as a reminder that work is not an exact differential (Section 2 .3), and so the value of its integral depends on the path.
2
2 2
ext
ext
( ) Work is done on a system by the surroundings. In this case the stops are pulled out, and the system is compressed to a new equilibrium state. ( ) Work is done on the surroundings by the system.
When the stops are pulled out, the system expands to a new equilibrium state.
Force
Work
The convention on is that it is positive when work is done on the system of interest and negative when the system does work on the surroundings.