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

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(A) The change in internal energy $\Delta U$ for an ideal gas is given by $\Delta U = n C_v \Delta T$.Using the relation $C_v = \frac{R}{\gamma-1}$,we

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

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(A) The change in internal energy $\Delta U$ for an ideal gas is given by $\Delta U = n C_v \Delta T$.Using the relation $C_v = \frac{R}{\gamma-1}$,we get $\Delta U = n \left( \frac{R}{\gamma-1} \right) \Delta T$.From the ideal gas law $PV = nRT$,at constant pressure $P$,we have $P \Delta V = nR \Delta T$.Substituting this into the internal energy equation: $\Delta U = \frac{P \Delta V}{\gamma-1}$.Given that the volume changes from $V$ to $2V$,the change in volume is $\Delta V = 2V - V = V$.Therefore,$\Delta U = \frac{P(V)}{\gamma-1} = \frac{PV}{\gamma-1}$.

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 $C_p$ to $C_v$):

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