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Engineering › Thermodynamics ›

Variations in Entropy for a Perfect Gas

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Describes the changes in Entropy which occur when a Perfect Gas is subjected to both reversible and irreversible operations

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Contents

The Variations Of Entropy For A Perfect Gas,
Entropy For An Irreversible Operation
Sketching Operations On P.v. And T.\f$\phi\f$ Graphs
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The Variations Of Entropy For A Perfect Gas,

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

13108/img_therm_14.jpg

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

At Constant Volume:-

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

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

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

At Constant Pressure

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

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

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

For an Isothermal Operation

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

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

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

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

Entropy For An Irreversible Operation

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

13108/img_therm_15.jpg

Stage 1

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

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

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

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

NOTE

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

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

13108/img_therm16.jpg

Constant Volume. 1 - 2

$P\;V^n\;=\;Constant$

$V\;=\;Constant$

$n\;=\;infinity$

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

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

Constant Pressure 1 - 3

$P\;V^n\;=\;Constant$

$P\;=\;Constant$

$n\;=\;0$

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

Isothermal 1 - 4

$P\;V^n\;=\;Constant$

$T\;=\;Constant$

$\therefore\;\;\;P\;V\;=\;Constant$

$n\;=\;1$

Adiabatic 1 - 5

$n\;=\;\gamma$

$\Phi\;=\;Constant$

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Last Modified: 2008-03-13 13:10:39 Page Rendered: 2010-09-24 12:57:00