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This section exmines the Comression and Expansion of a perfect Gas. The equations for AdiabatiC and Isothermal exoansions are derived

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Expansion and Compression of Gases

Invariably Gases follow Polytropic expansions

$PV^n\;=\;Constant$

(1)

$\frac{PV}{T}\;=\;Constant$

(2)

Divide equation 1 by 2

$PV^n\times \frac{T}{PV}\;=\;Constant$

(3)

$\therefore\;\;\;\;\;TV^{n-1}\;=\;Constant$

(4)

From equation 2

$V\;=\;Constant\times \frac{T}{P}$

(5)

Substituting in equation 1

$\frac{PT^n}{P^n}\;=\;Constant$

(6)

$\therefore\;\;\;\;\;\frac{T}{P^\frac{n-1}{n}}\;=\;Constant$

(7)

NOTE n is called the index of Compression.

Work Done during a Polytropic Compression.

Work done = $\int_{V_1}^{V_2}P\.dV$

But $PV^n\;=\;Constant$

$\therefore\;\;\;\;\;P\;=\;\frac{Constant}{V^n}$

(8)

$\therefore\;\;\;\;\;Work\;Done\;=\;Constant\;\int_{V_1}^{V_2}V^{-n}\;dV$

(9)

$\therefore\;\;\;\;\;Work\;Done\;=\;Constant\;\left[\frac{V^{1-n}}{1-n} \right]_{V_1}^{V_2}$

(10)

$=\;\left(\frac{V_2^{1-n}\;-\;V_1^{1-n}}{1\;-\;n} \right)\times Constant$

(11)

$=\;\frac{P_2V_2^n\times V_2^{1-n}\;-\;P_1V_1^n\times V_1^{1-n}}{1-n}$

(12)

$\therefore\;\;\;\;\;Work\;Done\;for\;an\;expansion\;=\;\frac{P_2V_2\;-\;P_1V_1}{1-n}$

(13)

$\therefore\;\;\;\;\;Work\;Done\;for\;an\;compression\;=\;\frac{P_1V_1\;-\;P_2V_2}{1-n}$

(14)

$=\;\frac{wR(T_1\;-\;T_2)}{n\;-\;1}$

(15)

The Heat Supplied during a Polytropic Expansion.

$q\;=\;\delta\,U\;+\;w$

(16)

$=\;\frac{w\;(T_1\;-\;T_2)}{n-1}\;+\;w\;C_V(T_2\;-\;T_1)$

(17)

$=\;(T_1\;-\;T_2)\left(\frac{w\;R}{n\;-\;1}\;-\;w\;C_V \right)$

(18)

$But\;\;\;\;\;R\;=\;(\gamma \;-\;1)C_V$

(19)

$\therefore\;\;\;\;\;q\;=\;R\;w(T_1\;-\;T_2)\left(\frac{1}{n\;-\;1}\;-\;\frac{1}{\gamma \;-\;1} \right)$

(20)

$\therefore\;\;\;\;\;q\;=\;R\;w(T_1\;-\;T_2)\frac{(\gamma \;-\;n)}{(n\;-\;1)(\gamma \;-\;1)}$

(21)

$Thus\;\;\;\;\;q\;=\;\left(\frac{\gamma \;-\;n}{\gamma \;-\;1} \right)\times Work\;Done$

(22)

The Adiabatic Process.

In and Adiabatic Compression or expansion the heat supplied is zero.

$\left(\frac{\gamma \;-\;n}{n \;-\;1} \right)\;=\;0$

(23)

Also for a small change $PdV\;=\;-\;\delta\;U\;\;\;\;\;\;\;(The\; First\; Law)$

$\therefore\;\;\;\;\;P\,dV\;=\;-\;w\;C_V\.dt$

(24)

$But\;\;\;\;\;\;P\;=\;\frac{w\;R\;T}{V}$

(25)

$\frac{w\;R\;T}{V}dV\;=\;-\;wC_V\;dt$

(26)

$\therefore\;\;\;\;\;R\frac{dV}{V}\;=\;-\;C_V\;\frac{dt}{T}$

(27)

Integrating:-

$R\;Ln\;V\;+\;C_V\;Ln\;T\;=\;Constant$

(28)

$\therefore\;\;\;\;\;LnV^R\times T^{C_V}\;=\;Constant$

(29)

Taking Anti-Logs;-

$V^R\times T^{C_V}\;=\;Constant$

(30)

$T\times V^{\frac{R}{C_V}}\;=\;Constant$

(31)

$T\times V^{\gamma \;-\;1}\;=\;Constant$

(32)

Thus for an Adiabatic Expansion $\;n\;=\;\gamma$

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$Work\;Done\;=\;\frac{wR(T_1\;-\;T_2)}{\gamma \;-\;1}\;=\;wC_V(T_1\;-\;T_2)\;=\;\delta \;U$

(33)

An Isothermal Compression or Expansion.

$P\times V\;=\;Constant$

(34)

i.e. The Index of Compression "n" = 1

$Work\;Done\;=\;\int_{V_1}^{V_2}P\;dV\;=\;\int_{V_1}^{V_2}n\frac{RT}{V}\;dV$

(35)

Integrating:-

$Work\;Done\;=\;n\;R\;T\;Ln\frac{V_2}{V_1}$

(36)

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$But\;\;\;\;\;q\;=\;\delta\;U\;+\;w$

(37)

$\delta\;U\;=\;0$

(38)

$\therefore\;\;\;\;\;Work \;Done\;=\;Heat\;Supplied$

(39)