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Calculate the work required to be done t...

Calculate the work required to be done to increase the temperature of 1 mole ideal gas by `30^@C`. Expansion of the gas takes place according to the relation `V prop T^(2/3)`. Take R=8.3 `"J mol"^(-1) K^(-1)`.

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`mu="1 mol" DeltaT=30^@ C , R=8.3 "J mol"^(-1) K^(-1)`
For 1 mol ideal gas PV=RT
`therefore P=(RT)/V`…(1)
Now, `V prop T^(2/3)` (given)
`therefore V=KT^(2/3)` K=constant ….(2)
From equation (1) and (2),
`P=(RT)/(KT^(2/3))`...(3)
Taking derivative of equation (2)
`dV=K(2/3T^((-1)/3)dT)`....(4)
Work to be done to increase temperature of ideal gas by `DeltaT`.
`W=int_T^(T+DeltaT) PdV`.....(5)
Putting value of P and dV from equation (3) and (4) ,
`W=int_T^(T+DeltaT) (RT)/(KT^(2/3)). 2/3 KT^((-1)/3)dT=2/3 R int_(T)^(T+DeltaT) dT`
`=2/3R[T]_T^(T+DeltaT)`
`W=2/3R[T+DeltaT-T]=2/3RDeltaT`
`=2/3xx8.3xx30`
`therefore` W=166 J
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