Chapter 14, Problem 14.6QP

 (a)

Interpretation Introduction

Interpretation: In accordance with the given conditions the number of possible arrangements and the entropy for the given set up has to be calculated.

Concept Introduction:

A thermodynamic system can have degenerate and non-degenerate energy levels.  There can be different possible arrangements of the particles in the various energy levels. These possible arrangements are defined as thermodynamic probability $\text{(W)}$ or microstates.  The thermodynamic probability can be calculated using the following formula.

${\text{W = g}}^{\text{N}}$

Where

             $\text{'N'}$’ is the total number of particles

             $\text{'g'}$ is the degeneracy of the energy level

The entropy and thermodynamic probability is related by Boltzmann equation.  As the number of possible arrangements increases the entropy also increases.

$\text{S = klnW}$

Where, $\text{S}$ is the entropy

            $\text{k}$ is the Boltzmann constant $\text{k=1}{\text{.38×10}}^{\text{-23}}{\text{JK}}^{\text{-1}}$

            $\text{W}$ is the thermodynamic probability

Answer to Problem 14.6QP

                The number of possible arrangements of the system with barrier $\text{(W) = 1024}$

    The entropy for the given system, $\text{S = 9}\text{.57}\times {\text{10}}^{-23}{\text{JK}}^{-1}$

   The number of possible arrangement of the system without barrier $\text{(W) = 1}\text{.05}\times {\text{10}}^6$

   The entropy for the given system, $\text{S = 1}\text{.91}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

Explanation of Solution

To record the given data

The number of particles in the system, $\text{N}=10$

The degeneracy of the system with the barrier, $g=2$

The degeneracy of the system without the barrier, $g=4$

To calculate the probability of arrangements of particles in the system with barrier

The probability of arrangements of  the particles in the system with barrier is 1024

There are ten  particles in the system.  With the barrier there are two cells in the system.  That is degeneracy is two.  On plugging in the values of $\text{N}$ and $\text{g}$ in the given equation we can calculate $\text{W}$.

${\text{W = g}}^{\text{N}}$

     $=2^{10}$

      $=1024$

Explanation:

To calculate the entropy of the given system with the barrier

Entropy of the system with barrier is found to be, $\text{S = 9}\text{.57}\times {\text{10}}^{-23}{\text{JK}}^{-1}$

The entropy of the system is calculated by plugging in the values of $\text{k}$ and $\text{W}$ in the following equation.

$\text{S = klnW}$

${\text{S = k ln g}}^{\text{N}}$

   $=\text{Nklng}$

   $\text{= (10)(1}\text{.38}\times {\text{10}}^{-23}J{K}^{-1})ln(2)$

    $\text{= 9}\text{.57}\times {\text{10}}^{-23}{\text{JK}}^{-1}$

Explanation:

To calculate the probability of arrangements of particles in the system without barrier

The probability of arrangement of particles in the system without barrier is $\text{ 1}\text{.05}\times {\text{10}}^6$

There are 10 particles in the system. Without the barrier there are four cells in the system.  That is degeneracy is four.   On plugging in the values of $\text{N}$ and $\text{g}$ in the given equation we can calculate $\text{W}$.

${\text{W = g}}^{\text{N}}$

     $=4^{10}$

     $\text{= 1}\text{.05}\times {\text{10}}^6$

Explanation:

To calculate the entropy of the given system without the barrier

Entropy of the system with barrier is found to be, $\text{S = 1}\text{.91}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

The entropy of the system is calculated by plugging in the values of $\text{k}$ and $\text{W}$ in the following equation.

$\text{S = klnW}$

${\text{S = k ln g}}^{\text{N}}$

   $=\text{Nklng}$

   $\text{= (10)(1}\text{.38}\times {\text{10}}^{-23}J{K}^{-1})ln(4)$

    $\text{= 1}\text{.91}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

Conclusion

The number of possible arrangements and the entropy for the given setup has been calculated in accordance with the given conditions.

(b)

Interpretation Introduction

Interpretation: In accordance with the given conditions the number of possible arrangements and the entropy for the given set up has to be calculated.

Concept Introduction:

A thermodynamic system can have degenerate and non degenerate energy levels.  There can be different possible arrangements of the particles in the various energy levels. These possible arrangements are defined as thermodynamic probability $\text{(W)}$ or microstates.  The thermodynamic probability can be calculated using the following formula.

${\text{W = g}}^{\text{N}}$

Where

             $\text{'N'}$’ is the total number of particles

             $\text{'g'}$ is the degeneracy of the energy level

The entropy and thermodynamic probability is related by Boltzmann equation.  As the number of possible arrangements increases the entropy also increases.

$\text{S = klnW}$

Where, $\text{S}$ is the entropy

            $\text{k}$ is the Boltzmann constant $\text{k=1}{\text{.38×10}}^{\text{-23}}{\text{JK}}^{\text{-1}}$

            $\text{W}$ is the thermodynamic probability

Answer to Problem 14.6QP

       (b)

  The number of possible arrangements of the system with barrier $\text{(W) = 1}\text{.13}\times {\text{10}}^{15}$

  The entropy for the given system, $\text{S = 4}\text{.78}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

  The number of possible arrangement of the system without barrier $\text{(W) = 1}\text{.27}\times {\text{10}}^{30}$

  The entropy for the given system, $\text{S = 9}\text{.57}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

Explanation of Solution

To record the given data

The number of particles in the system, $\text{N}=50$

The degeneracy of the system with the barrier, $g=2$

The degeneracy of the system without the barrier, $g=4$

To calculate the probability of arrangement of particles in the system with barrier

The probability of arrangement of particles in the system with barrier is $\text{ 1}\text{.13}\times {\text{10}}^{15}$

There are fifty  particles in the system.  With the barrier there are two cells in the system.  That is degeneracy is two.  On plugging in the values of $\text{N}$ and $\text{g}$ in the given equation we can calculate $\text{W}$

${\text{W = g}}^{\text{N}}$

     $=2^{50}$

     $\text{ = 1}\text{.05}\times {\text{10}}^6$

To calculate the entropy of the given system with the barrier

Entropy of the system with barrier is found to be, $\text{S = 4}\text{.78}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

The entropy of the system is calculated by plugging in the values of $\text{k}$ and $\text{W}$ in the following equation.

$\text{S = klnW}$

${\text{S = k ln g}}^{\text{N}}$

   $=\text{Nklng}$

   $\text{= (50)(1}\text{.38}\times {\text{10}}^{-23}J{K}^{-1})ln(2)$

    $\text{= 4}\text{.78}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

To calculate the probability of arrangement of particles in the system without barrier

The probability of arrangement of particles in the system without barrier is $\text{ 1}\text{.27}\times {\text{10}}^{30}$

There are fifty  particles in the system.  Without the barrier there are four cells in the system.  That is degeneracy is four.  On plugging in the values of $\text{N}$ and $\text{g}$ in the given equation we can calculate $\text{W}$.

${\text{W = g}}^{\text{N}}$

     $=4^{50}$

    $\text{ = 1}\text{.27}\times {\text{10}}^{30}$

To calculate the entropy of the given system without the barrier

Entropy of the system with barrier is found to be, $\text{S = 9}\text{.57}\times {\text{10}}^{-23}{\text{JK}}^{-1}$

The entropy of the system is calculated by plugging in the values of $\text{k}$ and $\text{W}$ in the following equation.

$\text{S = klnW}$

${\text{S = k ln g}}^{\text{N}}$

   $=\text{Nklng}$

   $\text{= (50)(1}\text{.38}\times {\text{10}}^{-23}J{K}^{-1})ln(4)$

  $\text{ = 9}\text{.57}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

Conclusion

The number of possible arrangements and the entropy for the given setup has been calculated in accordance with the given conditions.

(c)

Interpretation Introduction

Interpretation: In accordance with the given conditions the number of possible arrangements and the entropy for the given set up has to be calculated.

Concept Introduction:

A thermodynamic system can have degenerate and non degenerate energy levels.  There can be different possible arrangements of the particles in the various energy levels. These possible arrangements are defined as thermodynamic probability $\text{(W)}$ or microstates.  The thermodynamic probability can be calculated using the following formula.

${\text{W = g}}^{\text{N}}$

Where

             $\text{'N'}$’ is the total number of particles

             $\text{'g'}$ is the degeneracy of the energy level

The entropy and thermodynamic probability is related by Boltzmann equation.  As the number of possible arrangements increases the entropy also increases.

$\text{S = klnW}$

Where, $\text{S}$ is the entropy

            $\text{k}$ is the Boltzmann constant $\text{k=1}{\text{.38×10}}^{\text{-23}}{\text{JK}}^{\text{-1}}$

            $\text{W}$ is the thermodynamic probability

Answer to Problem 14.6QP

       (c)

 The number of possible arrangements of the system with barrier $\text{(W) = 1}\text{.27}\times {\text{10}}^{30}$

 The entropy for the given system, $\text{S = 9}\text{.57}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

 The number of possible arrangement of the system without barrier $\text{(W) = 1}\text{.61}\times {\text{10}}^{60}$

            The entropy for the given system, $\text{S = 1}\text{.91}\times {\text{10}}^{-21}{\text{JK}}^{-1}$

Explanation of Solution

To record the given data

The number of particles in the system, $\text{N}=100$

The degeneracy of the system with the barrier, $g=2$

The degeneracy of the system without the barrier, $g=4$

To calculate the probability of arrangement of particles in the system with barrier

The probability of arrangement of particles in the system with barrier is $\text{ 1}\text{.27}\times {\text{10}}^{30}$

There are hundred particles in the system.  With the barrier there are two cells in the system.  That is degeneracy is two.  On plugging in the values of $\text{N}$ and $\text{g}$ in the given equation we can calculate $\text{W}$.

${\text{W = g}}^{\text{N}}$

     $=2^{100}$

     $\text{ = 1}\text{.27}\times {\text{10}}^{30}$

To calculate the entropy of the given system with the barrier

Entropy of the system with barrier is found to be, $\text{S = 9}\text{.57}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

The entropy of the system is calculated by plugging in the values of $\text{k}$ and $\text{W}$ in the following equation.

$\text{S = klnW}$

${\text{S = k ln g}}^{\text{N}}$

   $=\text{Nklng}$

   $\text{= (100)(1}\text{.38}\times {\text{10}}^{-23}J{K}^{-1})ln(2)$

    $\text{= 9}\text{.57}\times {\text{10}}^{-22}{\text{JK}}^{-1}$

To calculate the probability of arrangement of particles in the system without barrier

The probability of arrangement of particles in the system without barrier is $\text{ 1}\text{.61}\times {\text{10}}^{30}$

There are hundred  particles in the system.  Without the barrier there are four cells in the system.  That is degeneracy is four.  On plugging in the values of $\text{N}$ and $\text{g}$ in the given equation we can calculate $\text{W}$.

${\text{W = g}}^{\text{N}}$

     $=4^{100}$

    $\text{ = 1}\text{.61}\times {\text{10}}^{30}$

To calculate the entropy of the given system without the barrier

Entropy of the system with barrier is found to be, $\text{S = 1}\text{.91}\times {\text{10}}^{-21}{\text{JK}}^{-1}$

The entropy of the system is calculated by plugging in the values of $\text{k}$ and $\text{W}$ in the following equation.

$\text{S = klnW}$

${\text{S = k ln g}}^{\text{N}}$

   $=\text{Nklng}$

  $\text{= (100)(1}\text{.38}\times {\text{10}}^{-23}J{K}^{-1})ln(4)$

  $\text{ = 1}\text{.91}\times {\text{10}}^{-21}{\text{JK}}^{-1}$

Conclusion

The number of possible arrangements and the entropy for the given setup has been calculated in accordance with the given conditions.