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For two thermo dynamic process temperature and volume are varied as follows. In first process, it is straight line having initial and final co-ordinates as $(V_{0}, T_{0})$ and $(2 V_{0}, 2 T_{0}) .$ Where as in second process it is a rectangular hyperbola having initial and final co-ordinates $(V_{0}, T_{0})$ and $(2 V_{0}, \frac{T_{0}}{2}) .$ Then the ratio of work done in first & second process is ______

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1.0

2.0

0.5

4.0

answer is C.

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Detailed Solution

In first process $P$ constant $W_{1} = n R (2 T_{0} - T_{0}) = n R T_{0}$

In second process $W_{2} = \int P d V = \int \frac{n R T}{V} d V$

$V T = c o n s \tan t (c) W = c n R \int_{V_{0}}^{2 V_{0}} \frac{d V}{V^{2}} = n R c {(\frac{- 1}{V})}_{V o}^{2 V_{0}}$

$W_{2} = n R C (\frac{1}{2 V o}) = \frac{n R T o}{2}$

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