A spherical balloon is filled with an ideal mono-atomic gas. At some instant its pressure was $P_{0}$ , volume $V_{0}$ and temperature $T_{0}$ . 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 ${6P}_{0} V_{0} [1- \frac{1}{\sqrt{n}}]$ when its volume increased to ${8V}_{0}$ . Find the value of $n$ . (assume ambient temperature to be close to 0 K)

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answer is 2.

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Detailed Solution

Rate of radiation = $σ e A T^{4}$ = constant
Or $r^{2} T^{4} = constant$
Or $\frac{4}{3} π r^{3} T^{6} = constant$
Or $V T^{6} = C$
Or $T V^{1 / 6} = constant$
(1)
Comparing it with polytropic equation $T V^{K - 1} = constant$
$W e g e t K - 1 = \frac{1}{6} \Rightarrow K = \frac{7}{6}$
Work done by gas $= \frac{n R (T_{1} - T_{2})}{K - 1} = \frac{n R}{(1 / 6)} (T_{0} - T_{2})$
$\begin{array}{l} = 6 n R (T_{0} - \frac{T_{0}}{\sqrt{2}}) \\ = 6 n R T_{0} (1 - \frac{1}{\sqrt{2}}) \end{array}} [T_{2} = \frac{T_{0}}{\sqrt{2}} a c c o r d i n g t o (1)] = 6 P_{0} V_{0} (1 - \frac{1}{\sqrt{2}}) S o n = 2$