Chapter 3, Problem 3A.1ST

Interpretation Introduction

Interpretation: The change in entropy when the pressure of a fixed amount of perfect gas is changed isothermally from $p_{\text{i}}$ to $p_{\text{f}}$ has to be calculated.  The origin of this change has to be stated.

Concept introduction: The degree of disorder or randomness is measured in term of entropy. According to thermodynamic definition, entropy is defined as the ratio of thermal energy to absolute temperature.

Answer to Problem 3A.1ST

The change in entropy when the pressure of a fixed amount of perfect gas for $\text{n}\text{mole}$ is changed isothermally from ${\text{p}}_{\text{i}}$ to ${\text{p}}_{\text{f}}$ is calculated as,

  $\text{Δ}S=n\text{Rln}\frac{P_{\text{i}}}{P_{\text{f}}}$

Explanation of Solution

The thermodynamic definition of entropy is expressed by formula,

  $\text{d}S=\frac{\text{d}{q}_{rev}}{T}$                                                                                                    $\left(1\right)$

Where,

• • $\text{d}S$ is the entropy change.
• • $\text{d}{q}_{\text{rev}}$ is the energy transfer at absolute temperature $\text{T}$.

Rearrange the above equation (1).

  $\text{d}{q}_{\text{rev}}=T\text{d}S$

From the first law of thermodynamics,

  $\text{d}U=\text{d}q-\text{d}W$                                                                                             $\left(2\right)$

Where,

• • $\text{d}U$ is the change in internal energy at constant volume.
• • $\text{d}W$ is the work done.
• • The change in internal energy at constant volume is expressed by the formula,

  $\text{d}U=C_V\text{d}T$ $\left(3\right)$

The work done is expressed as,

  $\text{d}W=P\text{d}v$ (4)

Substitute the values of $\text{d}{q}_{\text{rev}}$, $\text{d}U$ and $\text{d}W$ in equation (2).

  $\begin{array}{c}\text{d}U=\text{d}q-\text{d}W\\ \text{d}q=\text{d}U+\text{d}W\\ T\text{d}S=Cv\text{d}T+P\text{dv}\end{array}$

Here the process is isothermal.  It means that the temperature is constant.  Therefore, at constant temperature, $\text{d}T=\text{0}$ and the term, $C_{\text{V}}\text{d}T=\text{0}$.

Substitute, $C_{\text{V}}\text{d}T=\text{0}$ in the above equation.

  $\begin{array}{c}T\text{d}S=Cv\text{d}T+P\text{dv}\\ T\text{d}S=\text{0}+P\text{dv}\\ T\text{d}S=P\text{dv}\\ \text{d}S=\frac{P\text{dv}}{T}\end{array}$

Simplify the above equation.

  $T=\frac{P\text{dv}}{\text{d}S}$

From the ideal gas equation,

  $\begin{array}{c}PV=nRT\\ T=\frac{PV}{nR}\end{array}$

Substitute the value of $\text{T}$ in the above expression.

  $\begin{array}{c}T=\frac{PV}{nR}\\ \frac{P\text{d}v}{\text{d}S}=\frac{PV}{nR}\\ \text{d}S=\frac{nR\text{d}v}{V}\end{array}$

Integrate the above derivative between two states, initial and final, $i$ and $f$ respectively.

  $\begin{array}{c}{\displaystyle \underset{{\text{s}}_{\text{i}}}{\overset{{\text{s}}_{\text{f}}}{\int }}\text{d}S}={\displaystyle \underset{{\text{v}}_{\text{i}}}{\overset{{\text{v}}_{\text{f}}}{\int }}\frac{nR\text{dv}}{V}}\\ {\displaystyle \underset{{\text{s}}_{\text{i}}}{\overset{{\text{s}}_{\text{f}}}{\int }}\text{d}S}=nR{\displaystyle \underset{{\text{v}}_{\text{i}}}{\overset{{\text{v}}_{\text{f}}}{\int }}\frac{\text{dv}}{V}}\\ {\text{s}}_{\text{f}}-{\text{s}}_{\text{i}}=nR\text{ln}\left(\frac{V_{\text{f}}}{V_i}\right)\\ \Delta S=nR\text{ln}\left(\frac{V_{\text{f}}}{V_i}\right)\end{array}$

From ideal gas equation,

  $PV=nRT$

At constant temperature,

$PV$ will also be constant.  Therefore, it can be also written as,

  $\begin{array}{l}{P}_{\text{i}}{V}_{\text{i}}=P_{\text{f}}{V}_{\text{f}}\\ \frac{P_{\text{i}}}{P_{\text{f}}}=\frac{V_{\text{f}}}{V_{\text{i}}}\end{array}$

Replace $\frac{V_{\text{f}}}{V_{\text{i}}}$ by $\frac{P_{\text{i}}}{P_{\text{f}}}$ in the above expression.

  $\Delta S=nR\text{ln}\left(\frac{P_{\text{i}}}{P_{\text{f}}}\right)$

Hence, the entropy change when the pressure of a fixed amount of perfect gas is changed isothermally from ${\text{p}}_{\text{i}}$ to ${\text{p}}_{\text{f}}$ is shown by the expression,

  $\Delta S=nR\text{ln}\left(\frac{P_{\text{i}}}{P_{\text{f}}}\right)$