First slide

Law of equipartition of energy

Question

Suppose a diatomic gas gets ionised to a certain extent without any expenditure of heat energy. If the fractional change in the number of moles of the gas be $η$ , then ignoring any energy loss, the fractional change in the temperature of the gas will be

Moderate

A

$\frac{- η}{(η + 1)}$
B

$\frac{- η}{(η + 5)}$
C

$\frac{η}{(η + 1)}$
D

$\frac{η}{(η + 5)}$

Solution

Initially, E = $n_{o} (\frac{5}{2} RT) - - - - - - (i)$
Given $\frac{∆ n}{n_{o}} = η or, ∆ n = {ηn}_{o}$
Number of moles of $O_{2} left = (1 - n) n_{o}$
$∴ Finally, E = (1 - η) n_{o} (\frac{5 {RT}^{'}}{2}) + (2 {ηn}_{o}) (\frac{3 {RT}^{'}}{2}) - - - - - (ii)$
Equating equations (i) and (ii), $T^{'} = \frac{5 T}{5 + η}$
Fractional change in temperature = $\frac{T^{'}}{T} = 1 = \frac{- η}{η + 5}$

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