PHYSICAL CHEMISTRY-STUDENT SOLN.MAN.
PHYSICAL CHEMISTRY-STUDENT SOLN.MAN.
2nd Edition
ISBN: 9781285074788
Author: Ball
Publisher: CENGAGE L
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Chapter 19, Problem 19.27E
Interpretation Introduction

(a)

Interpretation:

The percentage of O2 molecules at 300K moving between 10m/s and 20m/s is to be calculated using Maxwell-Boltzmann distribution function.

Concept introduction:

The Maxwell-Boltzmann distribution is given by,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

This distribution depends on the mass of the particle and absolute temperature.

This probability distribution gives the distribution of velocities of any gas at a certain temperature.

Expert Solution
Check Mark

Answer to Problem 19.27E

The value of percentage of O2 molecules at 300K moving between 10m/s and 20m/s is 8.24×105.

Explanation of Solution

It is given that the O2 molecules at 300K move between 10m/s and 20m/s. The average velocity is 15m/s and dv is 10m/s. The mass of O2 molecule is 32g/mol.

The Maxwell-Boltzmann distribution function is,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

Substitute the values of mass of the system, average velocity, Boltzmann constant and temperature in the given formula.

G(v)dv=[4π(32×103kg/mol2π(8.314JK1mol1)(300K))3/2(15m/s)2×exp{32×103kg/mol(15m/s)22(8.314JK1mol1)(300K)}×10m/s]G(v)dv=[4π×2.9178(15m/s)2×exp{0.001443}×10m/s]G(v)dv=8.24×105

Thus, the value of percentage of O2 molecules at 300K moving between 10m/s and 20m/s is 8.24×105.

Conclusion

The value of percentage of O2 molecules at 300K moving between 10m/s and 20m/s is 8.28×105.

Interpretation Introduction

(b)

Interpretation:

The percentage of O2 molecules at 300K moving between 100m/s and 110m/s is to be calculated using Maxwell-Boltzmann distribution function.

Concept introduction:

The Maxwell-Boltzmann distribution is given by,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

This distribution depends on the mass of the particle and absolute temperature.

This probability distribution gives the distribution of velocities of any gas at a certain temperature.

Expert Solution
Check Mark

Answer to Problem 19.27E

The value of percentage of O2 molecules at 300K moving between 100m/s and 110m/s is 3.77×103.

Explanation of Solution

It is given that the O2 molecules at 300K move between 100m/s and 110m/s. The average velocity is 105m/s and dv is 10m/s. The mass of O2 molecule is 32g/mol.

The Maxwell-Boltzmann distribution function is,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

Substitute the values of mass of the system, average velocity, Boltzmann constant and temperature in the given formula.

G(v)dv=[4π(32×103kg/mol2π(8.314JK1mol1)(300K))3/2(105m/s)2×exp{32×103kg/mol(105m/s)22(8.314JK1mol1)(300K)}×10m/s]G(v)dv=[4π×2.9178(105m/s)2×exp{0.0707}×10m/s]G(v)dv=3.77×103

Thus, the value of percentage of O2 molecules at 300K moving between 100m/s and 110m/s is 3.77×103.

Conclusion

The value of percentage of O2 molecules at 300K moving between 100m/s and 110m/s is 3.77×103.

Interpretation Introduction

(c)

Interpretation:

The percentage of O2 molecules at 300K moving between 1000m/s and 1010m/s is to be calculated using Maxwell-Boltzmann distribution function.

Concept introduction:

The Maxwell-Boltzmann distribution is given by,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

This distribution depends on the mass of the particle and absolute temperature.

This probability distribution gives the distribution of velocities of any gas at a certain temperature.

Expert Solution
Check Mark

Answer to Problem 19.27E

The value of percentage of O2 molecules at 300K moving between 1000m/s and 1010m/s is 0.368.

Explanation of Solution

It is given that the O2 molecules at 300K move between 1000m/s and 1010m/s. The average velocity is 1005m/s and dv is 10m/s. The mass of O2 molecule is 32g/mol.

The Maxwell-Boltzmann distribution function is,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

Substitute the values of mass of the system, average velocity, Boltzmann constant and temperature in the given formula.

G(v)dv=[4π(32×103kg/mol2π(8.314JK1mol1)(300K))3/2(1005m/s)2×exp{32×103kg/mol(1005m/s)22(8.314JK1mol1)(300K)}×10m/s]G(v)dv=[4π×2.9178(1005m/s)2×exp{0.00645}×10m/s]G(v)dv=0.368

Thus, the value of percentage of O2 molecules at 300K moving between 1000m/s and 1010m/s is 0.368.

Conclusion

The value of percentage of O2 molecules at 300K moving between 1000m/s and 1010m/s is 0.368.

Interpretation Introduction

(d)

Interpretation:

The percentage of O2 molecules at 300K moving between 5000m/s and 5010m/s is to be calculated using Maxwell-Boltzmann distribution function.

Concept introduction:

The Maxwell-Boltzmann distribution is given by,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

This distribution depends on the mass of the particle and absolute temperature.

This probability distribution gives the distribution of velocities of any gas at a certain temperature.

Expert Solution
Check Mark

Answer to Problem 19.27E

The value of percentage of O2 molecules at 300K moving between 5000m/s and 5010m/s is 0.

Explanation of Solution

It is given that the O2 molecules at 300K move between 5000m/s and 5010m/s. The average velocity is 5005m/s and dv is 10m/s. The mass of O2 molecule is 32g/mol.

The Maxwell-Boltzmann distribution function is,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

Substitute the values of mass of the system, average velocity, Boltzmann constant and temperature in the given formula.

G(v)dv=[4π(32×103kg/mol2π(8.314JK1mol1)(300K))3/2(5005m/s)2×exp{32×103kg/mol(5005m/s)22(8.314JK1mol1)(300K)}×10m/s]

The value of exponential is negligible and approximately equal to zero thus, the value of percentage of O2 molecules at 300K moving between 5000m/s and 5010m/s is 0.

Conclusion

The value of percentage of O2 molecules at 300K moving between 5000m/s and 5010m/s is 0.

Interpretation Introduction

(e)

Interpretation:

The percentage of O2 molecules at 300K moving between 10000m/s and 10010m/s is to be calculated using Maxwell-Boltzmann distribution function. The explanation regarding distribution of velocities is to be stated.

Concept introduction:

The Maxwell-Boltzmann distribution is given by,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

This distribution depends on the mass of the particle and absolute temperature.

This probability distribution gives the distribution of velocities of any gas at a certain temperature.

Expert Solution
Check Mark

Answer to Problem 19.27E

The value of percentage of O2 molecules at 300K moving between 10000m/s and 10010m/s is 0.

Explanation of Solution

It is given that the O2 molecules at 300K move between 10000m/s and 10010m/s. The average velocity is 10005m/s and dv is 10m/s. The mass of O2 molecule is 32g/mol.

The Maxwell-Boltzmann distribution function is,

G(v)dv=4π(m2πkT)3/2v2emv2/2kTdv

Where,

m is the mass of the system.

v is the average velocity.

k is the Boltzmann constant.

T is the temperature.

Substitute the values of mass of the system, average velocity, Boltzmann constant and temperature in the given formula.

G(v)dv=[4π(32×103kg/mol2π(8.314JK1mol1)(300K))3/2(10005m/s)2×exp{32×103kg/mol(10005m/s)22(8.314JK1mol1)(300K)}×10m/s]

The value of exponential is negligible and approximately equal to zero thus, the value of percentage of O2 molecules at 300K moving between 10000m/s and 10010m/s is 0.

On comparison of the percentage of molecules moving with the given velocities it is observed that as the velocity increases, the percentage of molecules moving with that velocity increases till the interval of 1000m/s and 1100m/s. After this range of velocities, the percentage of molecules moving decreases with the increase in velocity.

Conclusion

The value of percentage of O2 molecules at 300K moving between 10000m/s and 10010m/s is 0.

As the velocity increases the percentage of molecules moving with that velocity increases till the interval of 1000m/s and 1100m/s. After this range of velocities, the percentage of molecules moving decreases with the increase in velocity.

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Chapter 19 Solutions

PHYSICAL CHEMISTRY-STUDENT SOLN.MAN.

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