Thermodynamic processes

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Question

The value of $γ = C_{P} / C_{V}$ is 4/3 for an adiabatic process of an ideal gas for which internal energy $U = K + nPV$ . The value of $n (K is a constant)$ is ___.

Moderate

Solution

${PV}^{γ} = constant$
${PV}^{4 / 3} = constant$
In $(V^{4}) + In (P^{3}) = constant$
$4 (\frac{dV}{V}) = - 3 (\frac{dP}{P})$
$4 P (dV) + 3 V (dP) = 0$
$3 (PdV + VdP) + PdV = 0$ …(i)
For adiabatic process, $dQ = 0 = dU + dW$
$= dU + PdV$ …(ii)
Compare Eqs. (i) and (ii), $dU = 3 (PdV + VdP)$
$U = K + 3 PV$
So, n = 3

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