The polytropic process for an ideal gas is given as $P V^{n} = constant$ . For this process, the expression for the work obtained is:
$W = \frac{P_{1} V_{1} [{(\frac{V_{2}}{V_{1}})}^{x} - 1]}{(y)}$
What is the value of $(x + y)$ ?

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Solution

$P V^{n} = c o n s t a n t = P_{1} V_{1}^{n} = P_{2} V_{2}^{n} = K (s a y)$
$∴ P = \frac{P}{V^{n}} \Rightarrow ω = - \int_{V_{1}}^{V_{2}} K . V^{- n} d V = \frac{K}{n - 1} {[V^{1 - n}]}_{V_{1}}^{1 - n}$
$= \frac{1}{n - 1} [\frac{K}{V_{2}^{n - 1}} - \frac{K}{V_{1}^{n - 1}}]$
$= \frac{P_{2} V_{2} - P_{1} V_{1}}{n - 1}$
$= \frac{P_{1} V_{1}}{n - 1} [\frac{P_{2} V_{2}}{P_{1} V_{1}} - 1]$
now, $P_{1} V_{1}^{n} = P_{2} V_{2}^{n}$
$\Rightarrow \frac{P_{2} V_{2}}{P_{1} V_{1}} = \frac{V_{1}^{n - 1}}{V_{2}^{n - 1}} = {(\frac{V_{2}}{V_{1}})}^{1 - n}$
$∴ ω = \frac{P_{1} V_{1}}{n - 1} [{(\frac{V_{2}}{V_{1}})}^{1 - n} - 1]$
$∴ x = 1 - n a n d y = n - 1$
$∴ x + y = z e r o .$

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