STANDARD MOLAR QUANTITIES OF A GAS

P URE S UBSTANCES IN S INGLE P HASES

7.9 STANDARD MOLAR QUANTITIES OF A GAS

Here B is the second virial coefficient, a function ofT. With this equation of state, Eq.

7.8.16 becomes

ln Bp

RT (7.8.18)

For a real gas at temperatureT and pressurep, Eq. 7.8.16 or 7.8.18 allows us to evaluate the fugacity coefficient from an experimental equation of state or a second virial coefficient.

We can then find the fugacity fromf Dp.

As we will see in Sec. 9.7, the dimensionless ratio D f =p is an example of an activity coefficientand the dimensionless ratiof =pıis an example of anactivity.

7.8.2 Liquids and solids

The dependence of the chemical potential on pressure at constant temperature is given by Eq. 7.8.5. With an approximation of zero compressibility, this becomes

ıCVm.p pı/ (7.8.19)

(pure liquid or solid, constantT)

7.9 STANDARD MOLAR QUANTITIES OF A GAS

Astandard molar quantityof a substance is the molar quantity in the standard state at the temperature of interest. We have seen (Sec. 7.7) that the standard state of a pureliquidor solid is a real state, so any standard molar quantity of a pure liquid or solid is simply the molar quantity evaluated at the standard pressure and the temperature of interest.

The standard state of agas, however, is a hypothetical state in which the gas behaves ideally at the standard pressure without influence of intermolecular forces. The properties of the gas in this standard state are those of an ideal gas. We would like to be able to relate molar properties of the real gas at a given temperature and pressure to the molar properties in the standard state at the same temperature.

We begin by using Eq. 7.8.7 to write an expression for the chemical potential of the real gas at pressurep0:

.p0/Dı(g)CRT lnf .p0/ pı Dı(g)CRT lnp0

pıCRTlnf .p0/

p0 (7.9.1)

We then substitute from Eq. 7.8.14 to obtain a relation between the chemical potential, the standard chemical potential, and measurable properties, all at the same temperature:

.p0/Dı(g)CRTln p0 pı C

Z p0 0

Vm RT p

dp (7.9.2)

(pure gas) Note that this expression for is not what we would obtain by simply integrating d D Vmdpfrompıtop0, because the real gas is not necessarily in its standard state of ideal-gas behavior at a pressure of1bar.

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.9 STANDARDMOLARQUANTITIES OF AGAS 186

Table 7.5 Real gases: expressions for differences between molar properties and standard molar values at the same temperature

General expression Equation of state

Difference at pressurep0 V DnRT =pCnB

Um Umı(g)

Z p0 0

Vm T @Vm

dpCRT p0Vm pTdB

dT Hm Hmı(g)

Z p0 0

Vm T @Vm

dp p

B TdB dT

Am Aım(g) RTln p0 pı C

Z p0 0

Vm RT p

dpCRT p0Vm RTln p pı Gm Gmı(g) RTln p0

pı C Z p0

RT p

dp RTln p

pıCBp Sm Smı(g) Rln p0

pı Z p0

@Vm

R p

dp Rln p

pı pdB dT Cp;m Cp;mı (g)

Z p0 0

T @2Vm

@T2

dp pTd2B

dT2

Recall that the chemical potentialof a pure substance is also its molar Gibbs energy Gm DG=n. The standard chemical potentialı(g) of the gas is the standard molar Gibbs energy,Gmı(g). Therefore Eq. 7.9.2 can be rewritten in the form

Gm.p0/DGmı(g)CRTln p0 pı C

Z p0 0

Vm RT p

dp (7.9.3)

The middle column of Table 7.5 contains an expression forGm.p0/ Gmı(g) taken from this equation. This expression contains all the information needed to find a relation between any other molar property and its standard molar value in terms of measurable properties. The way this can be done is as follows.

The relation between the chemical potential of a pure substance and its molar entropy is given by Eq. 7.8.3:

Sm D

(7.9.4) The standard molar entropy of the gas is found from Eq. 7.9.4 by changingtoı(g):

Smı(g)D

@ı(g)

(7.9.5) By substituting the expression for given by Eq. 7.9.2 into Eq. 7.9.4 and comparing the result with Eq. 7.9.5, we obtain

Sm.p0/DSmı(g) Rln p0 pı

Z p0 0

@Vm

R p

dp (7.9.6)

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.9 STANDARDMOLARQUANTITIES OF AGAS 187

The expression forSm Smı(g) in the middle column of Table 7.5 comes from this equation.

The equation, together with a value ofSmfor a real gas obtained by the calorimetric method described in Sec. 6.2.1, can be used to evaluateSmı(g).

Now we can use the expressions forGmandSmto find expressions for molar quantities such as Hm and Cp;m relative to the respective standard molar quantities. The general procedure for a molar quantityXmis to write an expression forXmas a function ofGmand Smand an analogous expression forXmı(g) as a function ofGmı(g) andSmı(g). Substitutions forGmandSm from Eqs. 7.9.3 and 7.9.6 are then made in the expression forXm, and the differenceXm Xmı(g) taken.

For example, the expression forUm Umı(g) in the middle column Table 7.5 was derived as follows. The equation defining the Gibbs energy,G D U T S CpV, was divided by the amountnand rearranged to

Um DGmCT Sm pVm (7.9.7)

The standard-state version of this relation is

Umı(g)DGmı(g)CT Smı(g) pıVmı(g) (7.9.8) where from the ideal gas lawpıVmı(g) can be replaced byRT. Substitutions from Eqs. 7.9.3 and 7.9.6 were made in Eq. 7.9.7 and the expression forUmı(g) in Eq. 7.9.8 was subtracted, resulting in the expression in the table.

For a real gas at low to moderate pressures, we can approximateVm by.RT =p/CB whereBis the second virial coefficient (Eq. 7.8.17). Equation 7.9.2 then becomes

ı(g)CRTln p

pı CBp (7.9.9)

The expressions in the last column of Table 7.5 use this equation of state. We can see what the expressions look like if the gas is ideal simply by settingBequal to zero. They show that when the pressure of an ideal gas increases at constant temperature,Gm andAm increase, Smdecreases, andUm,Hm, andCp;mare unaffected.

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

PROBLEMS 188

PROBLEMS

An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.

7.1 Derive the following relations from the definitions of˛,T, and:

˛D 1

T D 1

7.2 Use equations in this chapter to derive the following expressions for an ideal gas:

˛D1=T T D1=p 7.3 For a gas with the simple equation of state

VmD RT p CB

(Eq. 2.2.8), whereBis the second virial coefficient (a function ofT), find expressions for˛, T, and.@Um=@V /T in terms of dB=dT and other state functions.

7.4 Show that when the virial equationpVmDRT .1CBppCCpp2C /(Eq. 2.2.3) adequately represents the equation of state of a real gas, the Joule–Thomson coefficient is given by

JT D RT2ŒdBp=dTC.dCp=dT /pC Cp;m

Note that the limiting value at low pressure,RT2.dBp=dT /=Cp;m, is not necessarily equal to zero even though the equation of state approaches that of an ideal gas in this limit.

7.5 The quantity .@T =@V /U is called the Joule coefficient. James Joule attempted to evaluate this quantity by measuring the temperature change accompanying the expansion of air into a vacuum—the “Joule experiment.” Write an expression for the total differential ofU withT andV as independent variables, and by a procedure similar to that used in Sec. 7.5.2 show that the Joule coefficient is equal to

p ˛T =T CV

7.6 p–V–T data for several organic liquids were measured by Gibson and Loeffler.11The following formulas describe the results for aniline.

Molar volume as a function of temperature atpD1bar (298–358K):

VmDaCbT CcT2Cd T3 where the parameters have the values

aD69:287cm3mol 1 cD 1:044310 4cm3K 2mol 1 bD0:08852cm3K 1mol 1 d D1:94010 7cm3K 3mol 1 Molar volume as a function of pressure atT D298:15K (1–1000bar):

VmDe f ln.gCp=bar/

where the parameter values are

eD156:812cm3mol 1 f D8:5834cm3mol 1 gD2006:6

11Ref. [65].

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

PROBLEMS 189

(a) Use these formulas to evaluate˛,T,.@p=@T /V, and.@U=@V /T (the internal pressure) for aniline atT D298:15K andpD1:000bar.

(b) Estimate the pressure increase if the temperature of a fixed amount of aniline is increased by0:10K at constant volume.

7.7 (a) From the total differential ofHwithTandpas independent variables, derive the relation .@Cp;m=@p/T D T .@2Vm=@T2/p.

(b) Evaluate.@Cp;m=@p/T for liquid aniline at300:0K and1bar using data in Prob. 7.6.

7.8 (a) From the total differential ofV withT andpas independent variables, derive the relation .@˛=@p/T D .@T=@T /p.

(b) Use this relation to estimate the value of˛for benzene at25ıC and500bar, given that the value of˛is1:210 3K 1 at25ıC and1bar. (Use information from Fig. 7.2 on page 165.)

7.9 Certain equations of state supposed to be applicable to nonpolar liquids and gases are of the formp D Tf .Vm/ a=Vm2, wheref .Vm/is a function of the molar volume only andais a constant.

(a) Show that the van der Waals equation of state.pCa=Vm2/.Vm b/DRT (whereaand bare constants) is of this form.

(b) Show that any fluid with an equation of state of this form has an internal pressure equal toa=Vm2.

7.10 Suppose that the molar heat capacity at constant pressure of a substance has a temperature dependence given byCp;m D aCbT CcT2, wherea,b, andcare constants. Consider the heating of an amountnof the substance fromT1toT2at constant pressure. Find expressions forH andSfor this process in terms ofa,b,c,n,T1, andT2.

7.11 AtpD1atm, the molar heat capacity at constant pressure of aluminum is given by Cp;mDaCbT

where the constants have the values

aD20:67J K 1mol 1 bD0:01238J K 2mol 1

Calculate the quantity of electrical work needed to heat2:000mol of aluminum from300:00K to400:00K at1atm in an adiabatic enclosure.

7.12 The temperature dependence of the standard molar heat capacity of gaseous carbon dioxide in the temperature range298K–2000K is given by

Cp;mı DaCbT C c T2 where the constants have the values

aD44:2J K 1mol 1 bD8:810 3J K 2mol 1 cD 8:6105J K mol 1 Calculate the enthalpy and entropy changes when one mole of CO2 is heated at 1bar from 300:00K to800:00K. You can assume that at this pressureCp;mis practically equal toCp;mı . 7.13 This problem concerns gaseous carbon dioxide. At400K, the relation betweenp andVmat

pressures up to at least100bar is given to good accuracy by a virial equation of state truncated

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

PROBLEMS 190

at the second virial coefficient,B. In the temperature range300K–800K the dependence ofB on temperature is given by

BDa0Cb0T Cc0T2Cd0T3 where the constants have the values

a0D 521cm3mol 1 b0D2:08cm3K 1mol 1

c0D 2:8910 3cm3K 2mol 1 d0D1:39710 6cm3K 3mol 1

(a) From information in Prob. 7.12, calculate the standard molar heat capacity at constant pressure,Cp;mı , atT D400:0K.

(b) Estimate the value ofCp;munder the conditionsT D400:0K andpD100:0bar.

7.14 A chemist, needing to determine the specific heat capacity of a certain liquid but not having an electrically heated calorimeter at her disposal, used the following simple procedure known as drop calorimetry. She placed500:0g of the liquid in a thermally insulated container equipped with a lid and a thermometer. After recording the initial temperature of the liquid,24:80ıC, she removed a60:17-g block of aluminum metal from a boiling water bath at100:00ıC and quickly immersed it in the liquid in the container. After the contents of the container had become thermally equilibrated, she recorded a final temperature of27:92ıC. She calculated the specific heat capacityCp=mof the liquid from these data, making use of the molar mass of aluminum (M D 26:9815g mol 1) and the formula for the molar heat capacity of aluminum given in Prob. 7.11.

(a) From these data, find the specific heat capacity of the liquid under the assumption that its value does not vary with temperature. Hint: Treat the temperature equilibration process as adiabatic and isobaric (H D0), and equateHto the sum of the enthalpy changes in the two phases.

(b) Show that the value obtained in part (a) is actually an average value ofCp=mover the temperature range between the initial and final temperatures of the liquid given by

Z T2 T1

.Cp=m/dT T2 T1

7.15 Suppose a gas has the virial equation of statepVm D RT .1CBppCCpp2/, whereBpand Cpdepend only onT, and higher powers ofpcan be ignored.

(a) Derive an expression for the fugacity coefficient,, of this gas as a function ofp.

(b) For CO2(g) at0:00ıC, the virial coefficients have the valuesBp D 6:6710 3bar 1 andCp D 3:410 5bar 2. Evaluate the fugacityf at0:00ıC andpD20:0bar.

7.16 Table 7.6 on the next page lists values of the molar volume of gaseous H2O at400:00ıC and 12 pressures.

(a) Evaluate the fugacity coefficient and fugacity of H2O(g) at400:00ıC and200bar.

(b) Show that the second virial coefficientBin the virial equation of state,pVm DRT .1C B=VmCC =Vm2C /, is given by

BDRT lim

p!0

Vm RT

1 p

where the limit is taken at constantT. Then evaluateBfor H2O(g) at400:00ıC.

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

PROBLEMS 191

Table 7.6 Molar volume of H2O(g) at400:00ıCa

p=105Pa Vm=10 3m3mol 1 p=105Pa Vm=10 3m3mol 1

1 55:896 100 0:47575

10 5:5231 120 0:37976

20 2:7237 140 0:31020

40 1:3224 160 0:25699

60 0:85374 180 0:21447

80 0:61817 200 0:17918

abased on data in Ref. [70]

CHAPTER 8

THE SYSTEM, SURROUNDINGS, AND BOUNDARY

PHASES AND PHYSICAL STATES OF MATTER