Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Chapter 4, Problem 4.76E
Interpretation Introduction

Interpretation:

The value of ΔmixG as the mixture goes from xB=0 to xB=1.00 at 25.0°C is to be calculated. The graph between ΔmixG verses xB is to be plotted. The relative concentrations at which the value of ΔmixG is most negative is to be predicted.

Concept introduction:

The Gibb’s free energy of mixing determines whether the process of mixing is spontaneous or not at a constant pressure and temperature. The expression to calculate the Gibb’s free energy of mixing is shown below.

ΔmixG=nRT(xAlnxA+xBlnxB)

Expert Solution & Answer
Check Mark

Answer to Problem 4.76E

The graph between ΔmixG verses xB is shown below.

Physical Chemistry, Chapter 4, Problem 4.76E , additional homework tip  1

The value of ΔmixG is most negative when the relative concentration of B and A in terms of mole fraction is 0.5.

Explanation of Solution

The formula to calculate Gibb’s free energy of mixing is shown below.-

ΔmixG=nRT(xAlnxA+xBlnxB)…(1)

Where,

n is the total number of moles.

R is the gas constant.

T is the temperature.

xA and xB are the mole fractions of component AandB, respectively.

ΔmixGideal is the Gibbs energy of mixing for ideal mixture.

Conversion of temperature from Celsius to Kelvin is shown below.

T(K)=T(°C)+273K…(2)

Substitute the temperature (25.0°C) in the equation (2).

T=25.0°C+273T=298K

The temperature is 298K.

The mole fraction of gas A and gas B is equal to one as shown below.

xA+xB=1

Substitute the value of xB=0.1 in the above formula.

xA+0.1=1xA=0.9

The value of xA is 0.9.

Substitute the values of xA, xBR, T, and n in equation (1).

ΔmixG=(1mol)(8.314J/K×mol)(298K)[0.9ln(0.9)+0.1ln(0.1)]=2477.572[0.094820.23026]=805.409J

The value of ΔmixG at xB=0.1 is 805.409J.

Similarly, the value of ΔmixG as the mixture goes from xB=0 to xB=1.00 is calculated as follows:

xB       xA       lnxA       lnxB       xAlnxA       ΔmixG                                         +xBlnxB  

010000.1 0.9 0.105362.30259 0.32508  805.4090.20.80.223141.609440.50041239.780.30.70.356671.203970.610861513.46

0.4 0.6 0.510830.91629 0.67301   1667.430.50.50.693150.693150.693151717.320.60.40.916290.510830.673011667.430.70.31.203970.356670.610861513.46

0.8 0.2 1.609440.22314 0.5004   1239.780.90.12.302590.105360.32508805.40910000

Therefore, the value of ΔmixG is most negative when the relative concentration of B and A in terms of mole fraction is 0.5.

The graph between ΔmixG verses xB is shown below.

Physical Chemistry, Chapter 4, Problem 4.76E , additional homework tip  2

Figure 1

Conclusion

The graph between ΔmixG verses xB is shown in Figure 1. The value of ΔmixG is most negative when the relative concentration of B and A in terms of mole fraction is 0.5.

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Chapter 4 Solutions

Physical Chemistry

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