Atkins' Physical Chemistry
Atkins' Physical Chemistry
11th Edition
ISBN: 9780198769866
Author: ATKINS, P. W. (peter William), De Paula, Julio, Keeler, JAMES
Publisher: Oxford University Press
Question
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Chapter 13, Problem 13C.7P

(a)

Interpretation Introduction

Interpretation:

The electronic partition function of NO has to be calculated for the value of temperature from T=0 K to T=1000 K.  The graph of electronic partition function of NO and temperature has to be plotted.  The population of levels of NO molecule has to be calculated.

Concept introduction:

Statistical thermodynamics is used to describe all possible configurations in a system at given physical quantities such as pressure, temperature and number of particles in the system.  An important quantity in thermodynamics is partition function that is represented by the expression as shown below.

  q=igieεi/kT

(a)

Expert Solution
Check Mark

Answer to Problem 13C.7P

The electronic partition function of NO for the value of temperature from T=0 K to T=1000 K is shown below in Table.

Temperature (T)Partition (q)
0 K
200 K2.837095
400 K3.293905
600 K3.496046
800 K3.608667
1000 K3.680269

The graph of electronic partition function of NO and temperature is shown below.

Atkins' Physical Chemistry, Chapter 13, Problem 13C.7P , additional homework tip  1

The population of ground and first excited levels of NO molecule  is 0.6412_ and 0.3588_ respectively.

Explanation of Solution

The partition function in terms of wavenumber is given by the expression as shown below.

    q=jgjehcv˜jkT....................(1)

Where,

gj is the degeneracy of the ith level.

h is the Plank’s constant with a value 6.62608×1034J s.

c is the speed of light with value 2.997945×1010cms1.

v˜ is the wavenumber of the ith level.

k is the Boltzmann constant (1.381×1023J K1).

T is the temperature.

The degeneracy of the ground and first excited electronic level of NO molecule is 2.

The wavenumber of first excited electronic level of NO molecule is 121.1 cm1.

The wavenumber of ground electronic level of NO molecule is 0 cm1.

The temperature range is from T=0 K to T=1000 K.

The partition function of NO molecule is given by the expression as shown below.

    q=g0ehcv˜0kT+g1ehcv˜1kT

Substitute the values of h, c, v˜0, v˜1, g0, g1,and k in the above equation.

    q=((2)e(6.62608×1034J s)(2.997945×1010cms1)(0 cm1)(1.381×1023J K1)T+(2)e(6.62608×1034J s)(2.997945×1010cms1)(121.1 cm1)(1.381×1023J K1)T)q=(2)(1)+(2)e(174.193 KT)q=2+2e(174.193 KT)..............................(2)

Substitute the value T=0 K in the above expression.

    q=2+2e(174.193 K0 K)=2+=

Similarly, the partition is calculated for the temperature range from T=0 K to T=1000 K using the equation (1). The calculated values of partition function is shown below in the table.

Temperature (T)Partition (q)
0 K
200 K2.837095
400 K3.293905
600 K3.496046
800 K3.608667
1000 K3.680269

Therefore, the plot electronic partition function of NO and temperature is shown below.

Atkins' Physical Chemistry, Chapter 13, Problem 13C.7P , additional homework tip  2

Figure 1

The relative population of ground level of NO is given by the expression as shown below.

    N0N=g0ehcv˜0kT2+2e(174.193 KT)

Substitute the values of h, c, v˜0, g0, T=300 K,and k in the above equation.

    N0N=(2)e(6.62608×1034J s)(2.997945×1010cms1)(0 cm1)(1.381×1023J K1)(300 K)2+2e(174.193 K300 K)N0N=22+1.119N0N=0.6412

When the total population of the system is 1, the pollution of ground level of NO is calculated as shown below.

    N0(1)=0.6412N0=0.6412_

The pollution of first excited level of NO is calculated as shown below.

    N1=NN0

Where,

N1 is the population of first excited level of NO.

N is the total population of NO.

N0 is the population of ground level of NO.

Substitute the value of N0 and N in the above equation.

    N1=10.6412=0.3588_

Therefore, the population of ground and first excited levels of NO molecule  is 0.6412_ and 0.3588_ respectively.

(b)

Interpretation Introduction

Interpretation:

The mean electronic energy of NO at 300 K has to be calculated.

Concept introduction:

Refer to part (a).

(b)

Expert Solution
Check Mark

Answer to Problem 13C.7P

The mean electronic energy of NO at 300 K is 8.756×1022 J_.

Explanation of Solution

The degeneracy of the ground and first excited electronic level of NO molecule is 2.

The wavenumber of first excited electronic level of NO molecule is 121.1 cm1.

The wavenumber of ground electronic level of NO molecule is 0 cm1.

The temperature is 300 K.

The partition function of NO molecule is given by the expression as shown below.

    q=g0ehcβv˜0+g1ehcβv˜1

Where,

gj is the degeneracy of the ith level.

h is the Plank’s constant with a value 6.62608×1034J s.

c is the speed of light with value 2.997945×1010cms1.

v˜ is the wavenumber of the ith level.

β is the temperature dependent coefficient (1kT).

The mean energy, ε, is given by the expression as shown below.

    ε=(1qqβ)

Substitute the value of q in the above equation.

    ε=(1g0ehcβv˜0+g1ehcβv˜1(g0ehcβv˜0+g1ehcβv˜1)β)=(1g0ehcβv˜0+g1ehcβv˜1(hcv˜0g0ehcβv˜0hcv˜1g1ehcβv˜0))

Substitute the value of β=1kT in the above expression.

    ε=(1g0ehcv˜0kT+g1ehcv˜1kT(hcv˜0g0ehcv˜0kThcv˜1g1ehcv˜1kT))

Where,

k is the Boltzmann constant (1.381×1023J K1).

T is the temperature.

Substitute the values of h, c, v˜0, v˜1, g0, g1,and k in the above equation.

    ε=(1((2)e(6.62608×1034J s)(2.997945×1010cms1)(0 cm1)(1.381×1023J K1)(300 K)+(2)e(6.62608×1034J s)(2.997945×1010cms1)(121.1 cm1)(1.381×1023J K1)(300 K))(((6.62608×1034J s)(2.997945×1010cms1)(0 cm1)((2)e(6.62608×1034J s)(2.997945×1010cms1)(0 cm1)(1.381×1023J K1)(300 K)))((6.62608×1034J s)(2.997945×1010cms1)(121.1 cm1)(2)e(6.62608×1034J s)(2.997945×1010cms1)(121.1 cm1)(1.381×1023J K1)(300 K))))

The above expression can be further simplified as shown below.

    ε=(1(2+2e(0.5595))(0(4.811×1021×e(0.5595)))) J=4.811×1021(0.5715)3.14 J=8.756×1022 J_

Therefore, the mean electronic energy of NO at 300 K is 8.756×1022 J_.

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Chapter 13 Solutions

Atkins' Physical Chemistry

Ch. 13 - Prob. 13A.1AECh. 13 - Prob. 13A.1BECh. 13 - Prob. 13A.2AECh. 13 - Prob. 13A.2BECh. 13 - Prob. 13A.3AECh. 13 - Prob. 13A.3BECh. 13 - Prob. 13A.4AECh. 13 - Prob. 13A.4BECh. 13 - Prob. 13A.5AECh. 13 - Prob. 13A.5BECh. 13 - Prob. 13A.6AECh. 13 - Prob. 13A.6BECh. 13 - Prob. 13A.1PCh. 13 - Prob. 13A.2PCh. 13 - Prob. 13A.4PCh. 13 - Prob. 13A.5PCh. 13 - Prob. 13A.6PCh. 13 - Prob. 13A.7PCh. 13 - Prob. 13B.1DQCh. 13 - Prob. 13B.2DQCh. 13 - Prob. 13B.3DQCh. 13 - Prob. 13B.1AECh. 13 - Prob. 13B.1BECh. 13 - Prob. 13B.2AECh. 13 - Prob. 13B.2BECh. 13 - Prob. 13B.3AECh. 13 - Prob. 13B.3BECh. 13 - Prob. 13B.4AECh. 13 - Prob. 13B.4BECh. 13 - Prob. 13B.7AECh. 13 - Prob. 13B.7BECh. 13 - Prob. 13B.8AECh. 13 - Prob. 13B.8BECh. 13 - Prob. 13B.9AECh. 13 - Prob. 13B.9BECh. 13 - Prob. 13B.10AECh. 13 - Prob. 13B.10BECh. 13 - Prob. 13B.11AECh. 13 - Prob. 13B.11BECh. 13 - Prob. 13B.12AECh. 13 - Prob. 13B.12BECh. 13 - Prob. 13B.4PCh. 13 - Prob. 13B.5PCh. 13 - Prob. 13B.6PCh. 13 - Prob. 13B.7PCh. 13 - Prob. 13B.8PCh. 13 - Prob. 13B.10PCh. 13 - Prob. 13C.1DQCh. 13 - Prob. 13C.2DQCh. 13 - Prob. 13C.1AECh. 13 - Prob. 13C.1BECh. 13 - Prob. 13C.6AECh. 13 - Prob. 13C.6BECh. 13 - Prob. 13C.7AECh. 13 - Prob. 13C.7BECh. 13 - Prob. 13C.3PCh. 13 - Prob. 13C.7PCh. 13 - Prob. 13C.8PCh. 13 - Prob. 13C.9PCh. 13 - Prob. 13D.1DQCh. 13 - Prob. 13D.2DQCh. 13 - Prob. 13D.3DQCh. 13 - Prob. 13D.4DQCh. 13 - Prob. 13D.1AECh. 13 - Prob. 13D.1BECh. 13 - Prob. 13D.1PCh. 13 - Prob. 13D.2PCh. 13 - Prob. 13E.1DQCh. 13 - Prob. 13E.2DQCh. 13 - Prob. 13E.3DQCh. 13 - Prob. 13E.4DQCh. 13 - Prob. 13E.5DQCh. 13 - Prob. 13E.6DQCh. 13 - Prob. 13E.1AECh. 13 - Prob. 13E.1BECh. 13 - Prob. 13E.2AECh. 13 - Prob. 13E.2BECh. 13 - Prob. 13E.3AECh. 13 - Prob. 13E.3BECh. 13 - Prob. 13E.4AECh. 13 - Prob. 13E.4BECh. 13 - Prob. 13E.5AECh. 13 - Prob. 13E.5BECh. 13 - Prob. 13E.6AECh. 13 - Prob. 13E.6BECh. 13 - Prob. 13E.7AECh. 13 - Prob. 13E.7BECh. 13 - Prob. 13E.8AECh. 13 - Prob. 13E.8BECh. 13 - Prob. 13E.9AECh. 13 - Prob. 13E.9BECh. 13 - Prob. 13E.1PCh. 13 - Prob. 13E.2PCh. 13 - Prob. 13E.3PCh. 13 - Prob. 13E.4PCh. 13 - Prob. 13E.7PCh. 13 - Prob. 13E.9PCh. 13 - Prob. 13E.10PCh. 13 - Prob. 13E.11PCh. 13 - Prob. 13E.14PCh. 13 - Prob. 13E.15PCh. 13 - Prob. 13E.16PCh. 13 - Prob. 13E.17PCh. 13 - Prob. 13F.1DQCh. 13 - Prob. 13F.2DQCh. 13 - Prob. 13F.3DQCh. 13 - Prob. 13F.1AECh. 13 - Prob. 13F.1BECh. 13 - Prob. 13F.2AECh. 13 - Prob. 13F.2BECh. 13 - Prob. 13F.3AECh. 13 - Prob. 13F.3BECh. 13 - Prob. 13F.3PCh. 13 - Prob. 13F.4PCh. 13 - Prob. 13F.5PCh. 13 - Prob. 13F.6PCh. 13 - Prob. 13.1IA
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