First slide

Introduction-thermodynamics

Question

The maximum attainable temperature of an ideal gas in the process $P = P_{o} - α V^{2}$ (Where $P_{o}$ and $α$ are positive constants, n is a number of moles and R is universal gas constant) is

Difficult

A

$\frac{2 P_{o}}{3 n R} {(\frac{P_{o}}{3 α})}^{1 / 2}$
B

$\frac{P_{o}}{2 n R} {(\frac{2 P_{o}}{3 α})}^{1 / 2}$
C

$\frac{2 n R}{P_{o}} {(\frac{2 P_{o}}{3 α})}^{1 / 2}$
D

$\frac{2 P_{o}}{n R} {(\frac{P_{o}}{2 α})}^{1 / 2}$

Solution

Given $P = P_{o} - α V^{2}$
Where $P_{o}$ and $α$ are positive constants
From the ideal gas equation $P V = n R T$
$∴ (P_{o} - α V^{2}) V = n R T$
$\Rightarrow T = \frac{P_{o} V - α V^{3}}{n R}$
If T is max then $\frac{d T}{d V} = 0$
$0 = \frac{1}{n R} [P_{o} - 3 α V^{2}] \Rightarrow V = {(\frac{P_{o}}{3 α})}^{1 / 2}$
Corresponding pressure $P = P_{o} - α . \frac{P_{o}}{3 α} = \frac{2 P_{o}}{3}$
$\Rightarrow T = \frac{P V}{n R} = \frac{2 P_{o}}{3 R n} {(\frac{P_{o}}{3 α})}^{1 / 2}$

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