A spherical balloon is filled with an ideal mono-atomic gas. At some instant its pressure was $P_{0}$ , volume $V_{0}$ and temperature $T_{0}$ . If the balloon expanded in such a manner so that the rate of radiation from the balloon remained constant. It was found that the work done by gas inside the balloon was $6 P_{0} V_{0} [1 - \frac{1}{\sqrt{n}}]$ when its volume increased to 8 $V_{0}$ .Find the value of n. (Assume ambient temperature to be close to 0 K)

Difficult

Solution

Rate of radiation = $σ {eAT}^{4}$ = constant
or $r^{2} T^{4}$ = constant
or $\frac{4}{3} π r^{3} T^{6}$ = constant
or ${VT}^{6} = C$
or ${TV}^{1 / 6}$ = constant (1)
Comparing it with polytropic equation ${TV}^{K - 1}$ = constant
We get $K - 1 = \frac{1}{6}$
$\Rightarrow K = \frac{7}{6}$
Work done by gas = $\frac{nR (T_{1} - T_{2})}{K - 1} = \frac{nR}{(1 / 6)} (T_{0} - T_{2})$
$\begin{array}{l} = 6 nR (T_{0} - \frac{T_{0}}{\sqrt{2}}) \\ = 6 {nRT}_{0} (1 - \frac{1}{\sqrt{2}}) \end{array}\} [T_{2} = \frac{T_{0}}{\sqrt{2}} according to (1)]$
= $6 P_{0} V_{0} (1 - \frac{1}{\sqrt{2}})$
So n = 2

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