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
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Chapter 13, Problem 13B.10P
Interpretation Introduction

Interpretation:

The explicit summation vibrational partition function of I2 at 100K and 298K and the amount of I2 molecules in ground and first two excited levels at two temperature has to be stated.

Concept introduction:

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

  qigieβi

The vibrational partition function for molecules is represented as,

  qV=11eβhcν˜

Expert Solution & Answer
Check Mark

Answer to Problem 13B.10P

The explicit summation vibrational partition function of I2 at 100K is 4.049.

The explicit summation vibrational partition function of I2 at 298K is 4.737.

The ratio between ground and first two excited states at 100K is 9.76×105.

The ratio between ground and first two excited states at 298K is 0.045.

Explanation of Solution

The important quantity in the thermodynamic is partition function that is represented as shown below.

  qigieβi

Where

  • q is vibrational partition function.
  • gi is degeneracy.
  • β is Boltzmann constant.
  • ei is energy of system.

The vibrational partition function for molecules is represented as,

  qV=11eβhcν˜

The value of βhcν˜=hcν˜kT for temperature 100K is calculated as shown below.

The value of kThc=69.38cm1

Now substitute the value ν˜=0cm1. Then the value βhcν˜=hcν˜kT becomes:

  βhcν˜=hcν˜kT=069.38cm1=0

Now substitute the value ν˜=215.30cm1 . Then the value βhcν˜=hcν˜kT becomes:

  βhcν˜=hcν˜kT=215.30cm169.38cm1=3.103

Now substitute the value ν˜=425.39cm1. Then the value βhcν˜=hcν˜kT becomes:

  βhcν˜=hcν˜kT=425.39cm169.38cm1=6.131

Now substitute the value ν˜=636.27cm1. Then the value βhcν˜=hcν˜kT becomes:

  βhcν˜=hcν˜kT=636.27cm169.38cm1=9.17

Now substitute the value ν˜=845.93cm1. Then the value βhcν˜=hcν˜kT becomes

  βhcν˜=hcν˜kT=845.93cm169.38cm1=12.192

The partition function of iodine at 100K with 0cm1 energy is calculated as,

  qV=11ehcν˜kT=11e0=

The partition function of iodine at 100K with 215.30cm1 energy is calculated as shown below.

  qV=11ehcν˜kT=11e3.103=1.047

The partition function of iodine at 100K with 425.39cm1 energy is calculated as,

  qV=11ehcν˜kT=11e6.13=1.002

The partition function of iodine at 100K with 636.27cm1 energy is calculated as,

  qV=11ehcν˜kT=11e9.17=1.000

The partition function of iodine at 100K with 845.93cm1 energy is calculated as,

  qV=11ehcν˜kT=11e12.192=1.00

The summation of theses partition function at different energy levels neglecting value is calculated as shown below.

  qv=qiv=1.047+1.002+1.00+1.00=4.049

The value of βhcν˜=hcν˜kT for temperature 298K is calculated as shown below.

The value of kThc=207.244cm1

Now substitute the value ν˜=0cm1. Then the value βhcν˜=hcν˜kT becomes:

  βhcν˜=hcν˜kT=0207.244cm1=0

Now substitute the value ν˜=215.30cm1 . Then the value βhcν˜=hcν˜kT becomes

  βhcν˜=hcν˜kT=215.30cm1207.244cm1=1.038

Now substitute the value ν˜=425.39cm1. Then the value βhcν˜=hcν˜kT becomes

  βhcν˜=hcν˜kT=425.39cm1207.244cm1=2.052

Now substitute the value ν˜=636.27cm1. Then the value βhcν˜=hcν˜kT becomes:

  βhcν˜=hcν˜kT=636.27cm1207.244cm1=3.070

Now substitute the value ν˜=845.93cm1. Then the value βhcν˜=hcν˜kT becomes:

  βhcν˜=hcν˜kT=845.93cm1207.244cm1=4.081

The partition function of iodine at 298K with 0cm1 energy is calculated as shown below.

  qV=11ehcν˜kT=11e0=

The partition function of iodine at 298K with 215.30cm1 energy is calculated as shown below.

  qV=11ehcν˜kT=11e1.038=1.54

The partition function of iodine at 298K with 425.39cm1 energy is calculated as shown below.

  qV=11ehcν˜kT=11e2.052=1.14

The partition function of iodine at 298K with 636.27cm1 energy is calculated as shown below.

  qV=11ehcν˜kT=11e3.070=1.04

The partition function of iodine at 298K with 845.93cm1 energy is calculated as shown below.

  qV=11ehcν˜kT=11e4.081=1.017

The summation of theses partition function at different energy levels neglecting value is calculated as shown below.

  qV=qiV=1.54+1.14+1.04+1.017=4.737

The amount of iodine at ground and first two excited state is calculated by using Boltzmann distribution as,

  pipj=e(εjεikT)

Where,

  • pi,pj are probabilities states.
  • εi,εj are energy sates.
  • k is Boltzmann constant
  • T is Temperature

The ratio between ground and first two excited states at 100K is calculated as shown below.

  pgroundpexcited=e(εexcitedεgroundkT)=e(640.690kT)=e(640.69cm1×6.63×1034Js×3×1010cms-11.38×1023JK1×100K)=9.76×105

The ratio between ground and first two excited states at 298K is calculated as,

  pgroundpexcited=e(εexcitedεgroundkT)=e(640.690kT)=e(640.69cm1×6.63×1034Js×3×1010cm.s-11.38×1023J.K1×298K)=0.045

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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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