Q.A man starts bicycling in the morning at a temperature around 25°C; he checked the pressure of the tyre which was equal to 500 kPa. In the afternoon he found that the absolute pressure in the tyre had increased to 520 kPa. Assuming the expansion of the tyre is negligible, what is the temperature of the tyre in the afternoon?

In an isochoric (constant-volume) process, no work is done by or on the gas, so W = 0.

The work done by a gas during a small volume change is dW = P dV. In an isochoric process, the volume is held fixed, so dV = 0 throughout the process, which means

W = integral of P dV = 0

This is the defining mechanical consequence of constant volume: even though pressure and temperature can change (e.g., by heating the gas in a rigid, sealed container), and heat Q can flow in or out (going entirely into changing internal energy, by the first law: Q = delta U since W = 0), no work is exchanged with the surroundings simply because the boundary of the gas does not move.

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