A spherical balloon is filled with an ideal $$mono-atomic$$ gas. At some instant its pressure was $$P0$$, volume $$V0$$ and temperature $$T0$$. 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 $$6P0V01$$−$$1n$$ when its volume increased to $$8V0$$.Find the value of n. (Assume$$ ambient temperature to be close to $$0$$ $$K)

Question

A spherical balloon is filled with an ideal $$mono-atomic$$ gas. At some instant its pressure was $$P0$$, volume $$V0$$ and temperature $$T0$$. 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  $$6P0V01$$−$$1n$$ when its volume increased to $$8V0$$.Find the value of n. (Assume$$ ambient temperature to be close to $$0$$ $$K)

Infinity Learn · Accepted Answer

Rate of radiation $$=$$  σ$$eAT4=$$ constantor $$r2T4$$  $$=$$ constantor $$43$$$$π$$$$r3T6$$  $$=$$ constantor  $$VT6=Cor$$  $$TV1/6$$ $$=$$ constant      (1)Comparing$$ it with polytropic equation   TVK−$$1$$ $$=$$ constantWe get  K−$$1=16$$⇒ $$K=76$$   Work done by gas $$=$$ $$nRT1$$−$$T2K$$−$$1=nR(1/6)T0$$−$$T2$$ $$=6nRT0$$−$$T02=6nRT01$$−$$12T2=T02$$ according to $$(1)  $$=$$  $$6P0V01$$−$$12So$$ n $$=$$ $$2$$

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