derive an expression for work done in adiabatic process?

This derivation is well explained in the study material. 

Please refer the following link: http://cbse.meritnation.com/study-online/study-material/RjC3kc9dxHXETXK5FO19yQ!!/iuxrGaHwEgkX1NU6ZcQduA!!#5601

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 The mathematical equation for an ideal gas undergoing a reversible (i.e., no entropy generation) adiabatic process is

 P V^{gamma} = operatorname{constant} qquad

where P is pressure, V is specific or molar volume, and

 gamma = {C_{P} over C_{V}} = frac{f + 2}{f},

 C_{P}  being the specific heat for constant pressure,  C_{V}  being the specific heat for constant volume,  gamma  is the adiabatic index, and  f  is the number of degrees of freedom (3 for monatomic gas, 5 for diatomic gas).

For a monatomic ideal gas,  gamma = 5/3 ,, and for a diatomic gas (such as nitrogen and oxygen, the main components of air gamma = 7/5 ,.[4]Note that the above formula is only applicable to classical ideal gases and not Bose–Einstein or Fermi gases.

For reversible adiabatic processes, it is also true that

 P^{1-gamma}T^{gamma}= operatorname{constant}
 VT^{f/2} = operatorname{constant}

where T is an absolute temperature.

This can also be written as

 TV^{gamma - 1} = operatorname{constant}
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