Physics

Physics

Problem

Calculate the work done when one mole of a perfect gas is compressed adiabatically. The initial pressure and volume of the gas are 105 N/m210^5\,\text{N/m}^2 and 66 litres respectively. The final volume of the gas is 22 litre. Molar specific heat of the gas at constant volume is 3R/23R/2.
Intuitive explanation

In an adiabatic process, no heat flows in or out, so all the work done on the gas goes directly into changing its internal energy. The key is recognizing that for a perfect gas, ΔU=nCVΔT\Delta U = nC_V\Delta T, and you can find the temperature change using the adiabatic relation TVγ−1=constantTV^{\gamma-1} = \text{constant} (where γ\gamma depends on CVC_V). Once you know how much the internal energy changes, the first law tells you the work done—no need to integrate P dVP\,dV manually.

Intuitive explanation

In an adiabatic process, no heat flows in or out, so all the work done on the gas goes directly into changing its internal energy. The key is recognizing that for a perfect gas, ΔU=nCVΔT\Delta U = nC_V\Delta T, and you can find the temperature change using the adiabatic relation TVγ−1=constantTV^{\gamma-1} = \text{constant} (where γ\gamma depends on CVC_V). Once you know how much the internal energy changes, the first law tells you the work done—no need to integrate P dVP\,dV manually.

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