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An ideal gas whose adiabatic exponent (g...

An ideal gas whose adiabatic exponent `(gamma)` equal 1.5, expands so that the amount of heat transferred to it is equal to the decrease in its internal energy. Find the `T - V` equation for the process

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The work done by a gas in a polytropic process can be given as
`W=n(C-C_V)(T_2-T_1)` …(3.98)
Here initial temperature of gas is given as `T_0` and the final temperature can be obtained from equation-(3.97) for initial and final state of gas as
`T_1V_1^((gamma-1)/2) =T_2V_2^((gamma-1)/2)`
Here `T_1=T_0` and `V_2=etaV_1`, thus
`T_2=T_0(V_1/V_2)^((gamma-1)/2)=T_0eta^((1-gamma)/2)`
Now from equation-(3.98),work done is
`W=n(-C_V-C_V)(T_2-T_1)`
or `W=-(2R)/(gamma-1)[T_0 (eta)^((1-gamma)/2)-T_0]` [As n=1 mole]
or `W=(2RT_0)/(gamma-1)[1-(eta)^((1-gamma)/2)]`
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