Chapter 16, Problem 16.41SP

(a)

Interpretation Introduction

Interpretation:

Molecules of an ideal gas is distributed in equal volume among three bulbs A, B, and C.  The number of ways to achieve the state of 2 molecules in bulb A has to be calculated.  Using Boltzmann’s equation entropy of the state has to be calculated.

Concept Introduction:

Boltzmann proposed an equation to calculate the entropy of a system in a particular state by relating it to the number of ways that the state can be achieved.  The equation is given as,

$S=k\ln W$

Where,

$\begin{array}{l}S=\text{ Entropy of the state}\\ \ln \text{ W}\text{= natural logarithm of the number of ways the state can be achieved}\text{.}\\ \text{k}\text{= Boltzmann's constant}\end{array}$

(b)

Interpretation Introduction

Interpretation:

Molecules of an ideal gas is distributed in equal volume among three bulbs A, B, and C.  The number of ways to achieve the state of 2 molecules randomly distributed among bulbs A, B, and C has to be calculated.  Using Boltzmann’s equation entropy of the state has to be calculated.

Concept Introduction:

Boltzmann proposed an equation to calculate the entropy of a system in a particular state by relating it to the number of ways that the state can be achieved.  The equation is given as,

$S=k\ln W$

Where,

$\begin{array}{l}S=\text{ Entropy of the state}\\ \ln \text{ W}\text{= natural logarithm of the number of ways the state can be achieved}\text{.}\\ \text{k}\text{= Boltzmann's constant}\end{array}$

(c)

Interpretation Introduction

Interpretation:

Molecules of an ideal gas is distributed in equal volume among three bulbs A, B, and C.  The number of ways to achieve the state of 3 molecules occupying bulb A has to be calculated.  Using Boltzmann’s equation entropy of the state has to be calculated.

Concept Introduction:

Boltzmann proposed an equation to calculate the entropy of a system in a particular state by relating it to the number of ways that the state can be achieved.  The equation is given as,

$S=k\ln W$

Where,

$\begin{array}{l}S=\text{ Entropy of the state}\\ \ln \text{ W}\text{= natural logarithm of the number of ways the state can be achieved}\text{.}\\ \text{k}\text{= Boltzmann's constant}\end{array}$

(d)

Interpretation Introduction

Interpretation:

Molecules of an ideal gas is distributed in equal volume among three bulbs A, B, and C.  The number of ways to achieve the state of 3 molecules randomly distributed among bulbs A, B, and C has to be calculated.  Using Boltzmann’s equation entropy of the state has to be calculated.

Concept Introduction:

Boltzmann proposed an equation to calculate the entropy of a system in a particular state by relating it to the number of ways that the state can be achieved.  The equation is given as,

$S=k\ln W$

Where,

$\begin{array}{l}S=\text{ Entropy of the state}\\ \ln \text{ W}\text{= natural logarithm of the number of ways the state can be achieved}\text{.}\\ \text{k}\text{= Boltzmann's constant}\end{array}$

(e)

Interpretation Introduction

Interpretation:

Molecules of an ideal gas is distributed in equal volume among three bulbs A, B, and C.  The number of ways to achieve the state of 1 mol of molecules occupying bulb A has to be calculated.  Using Boltzmann’s equation entropy of the state has to be calculated.

Concept Introduction:

Boltzmann proposed an equation to calculate the entropy of a system in a particular state by relating it to the number of ways that the state can be achieved.  The equation is given as,

$S=k\ln W$

Where,

$\begin{array}{l}S=\text{ Entropy of the state}\\ \ln \text{ W}\text{= natural logarithm of the number of ways the state can be achieved}\text{.}\\ \text{k}\text{= Boltzmann's constant}\end{array}$

(f)

Interpretation Introduction

Interpretation:

Molecules of an ideal gas is distributed in equal volume among three bulbs A, B, and C.  The number of ways to achieve the state of 1 mol of molecules randomly distributed in bulb A, B, and C has to be calculated.  Using Boltzmann’s equation entropy of the state has to be calculated.

$\Delta S$ for the conversion of the state from (e) to (f) has to be determined and compared with that of the $\Delta S$ calculated by the formula $\Delta S=nR\ln \left(\frac{V_{final}}{V_{initial}}\right)$

Concept Introduction:

Boltzmann proposed an equation to calculate the entropy of a system in a particular state by relating it to the number of ways that the state can be achieved.  The equation is given as,

$S=k\ln W$

Where,

$\begin{array}{l}S=\text{ Entropy of the state}\\ \ln \text{ W}\text{= natural logarithm of the number of ways the state can be achieved}\text{.}\\ \text{k}\text{= Boltzmann's constant}\end{array}$