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# Variations in Entropy

Changes in Entropy that occur when an Ideal Gas is subjected to both reversible and irreversible operations
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## The Variations Of Entropy For A Perfect Gas,

$\delta&space;\Phi&space;&space;=&space;\frac{dq}{T}\;for\;a\;reversible\;operation$

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$\delta&space;\Phi&space;\;T&space;=&space;\delta\,q$

At Constant Volume:-

$\delta&space;\Phi&space;=&space;&space;\frac{w\,C_V\,dt}{T}$

$\therefore\;\;\;\;\;\Phi&space;_2&space;-&space;\Phi&space;_1&space;=&space;\int_{T_1}^{T_2}\frac{w\;C_V\,dt}{T}$

$=&space;wC_V\;Ln\frac{T_2}{T_1}\;\;\;\;\;or\;\;\;\;\;w\;C_V\;ln\frac{P_2}{P^1}$

At Constant Pressure

$\delta&space;\Phi&space;&space;=&space;\frac{w\;C_P\;dt}{T}$

$\therefore\;\;\;\;\;\Phi&space;_2&space;-&space;\Phi&space;_1&space;=&space;\int_{T_1}^{T_2}\;w\;C_P\;dt$

$=&space;w\;C_P\;Ln\frac{T_2}{T_1}\;\;\;\;\;or\;\;\;\;\;w\;C_P\;Ln\frac{V_2}{V_1}$

For an Isothermal Operation

$Heat\;Supplied&space;=&space;Work\;Done&space;+&space;\delta\,U\,(&space;Constant)$

$=&space;P\;\delta&space;\,V$

$\therefore\;\;\;\;\;\delta&space;\Phi&space;&space;=&space;\frac{p\delta&space;V}{T}&space;=&space;w\;R\;\frac{dv}{V}$

$Thus\;\;\;\;\;\Phi&space;_2&space;-&space;\Phi_1&space;=&space;w\,R\,Ln\frac{V_2}{V_1}\;\;\;\;\;or\;\;\;\;\;w\,R\,Ln\frac{P_1}{P_2}$

## Entropy For An Irreversible Operation

Consider a gas changing from $\inline&space;P_1V_1T_1\;to\;P_2V_2T_2$. Now since the Function of State depends upon the values of P:V:&T at 1 & 2 and is independent of the process.

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Stage 1

$\Phi&space;_a&space;-&space;\Phi&space;_1&space;=&space;wC_V\;Ln\;\frac{T_a}{T_1}\;\;\;\;\;or\;\;\;\;\;wC_V\;Ln\;\frac{P_2}{P_1}$

$\Phi&space;_2&space;-&space;\Phi&space;_a&space;=&space;wC_P\;Ln\;\frac{T_2}{T_a}\;\;\;\;\;or\;\;\;\;\;wC_V\;Ln\;\frac{V_2}{V_1}$

$\therefore\;\;\;\;\;\Phi&space;_2&space;-&space;\Phi&space;_1&space;=&space;w\left(&space;C_P\;Ln\;\frac{P_2}{P_1}&space;+&space;C_V\;Ln\;\frac{V_2}{V_1}&space;\right)$

## Sketching Operations On P.v. And T.$\phi$ Graphs

NOTE

The Area under the curve on a P.V. graph is the Work Done

The Area under the curve on a T.$\inline&space;\Phi$ graph is the heat Supplied

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Constant Volume. 1 - 2

$P\;V^n&space;=&space;Constant$
$V&space;=&space;Constant$
$n&space;=&space;infinity$

$\Phi&space;_2&space;-&space;\Phi&space;_2&space;=&space;w\;C_V\;Ln\;\frac{T_2}{T_1}$

$\therefore\;\;\;\;\;\Phi&space;&space;=&space;w\;C_V\;Ln\;T$

Constant Pressure 1 - 3

$P\;V^n&space;=&space;Constant$
$P&space;=&space;Constant$
$n&space;=&space;0$

$\Phi&space;_2&space;-&space;\Phi&space;_1&space;=&space;wC_P\;Ln\;\frac{T_2}{T_1}$

Isothermal 1 - 4

$P\;V^n&space;=&space;Constant$
$T&space;=&space;Constant$

$\therefore\;\;\;P\;V&space;=&space;Constant$
$n&space;=&space;1$

$n&space;=&space;\gamma$
$\Phi&space;=&space;Constant$