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Q.
For an adiabatic process, the relation between $U$ , $P$ and $V$ for an ideal gas is $U=2+3 \, PV$ . The gas is
NTA AbhyasNTA Abhyas 2022
Solution:
For an adiabatic process
$dQ=0=dU+dW$
or $0=dU+PdV$
From the given equation
$dU=3\left(\right.PdV+VdP\left.\right)$
$\therefore $ $0=3\left(\right.PdV+VdP\left.\right)+PdV$
or $4P\left(\right.dV\left.\right)+3V\left(\right.dP\left.\right)=0$
or $4 \left(\frac{\text{dV}}{\text{V}}\right) = - \left(\frac{\text{dP}}{\text{P}}\right)$
On integrating, we get
$ln\left(\right.V^{4}\left.\right)+ln\left(\right.P^{3}\left.\right)=$ constant
or $PV^{\frac{4}{3}}$ = constant
i.e., $\gamma = \frac{4}{3}$
i.e., gas is polyatomic.