The change in internal energy of a mass of ga | vedclass

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(A) The change in internal energy $\Delta U$ of an ideal gas is given by the formula $\Delta U = nC_v \Delta T$.From the ideal gas equation, $PV = nRT

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$\frac{PV}{\gamma - 1}$

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(A) The change in internal energy $\Delta U$ of an ideal gas is given by the formula $\Delta U = nC_v \Delta T$.From the ideal gas equation, $PV = nRT$, we have $n \Delta T = \frac{\Delta(PV)}{R}$.Since the pressure $P$ is constant, $\Delta(PV) = P(V_f - V_i) = P(2V - V) = PV$.Thus, $n \Delta T = \frac{PV}{R}$.We know that $C_v = \frac{R}{\gamma - 1}$.Substituting these into the internal energy formula: $\Delta U = n \left( \frac{R}{\gamma - 1} \right) \Delta T = \frac{R}{\gamma - 1} \left( \frac{PV}{R} \right) = \frac{PV}{\gamma - 1}$.

The change in internal energy of a mass of gas, when the volume changes from $V$ to $2V$ at constant pressure $P$ is (where $\gamma$ is the ratio of specific heat at constant pressure to that at constant volume, $\gamma = \frac{C_p}{C_v}$):

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