The value of $γ = \frac{C_{P}}{C_{V}}$ is $\frac{4}{3}$ for an adiabatic process of an ideal gas for which the internal energy is given by $d U = n (V d P + P d V)$ . The value of $n$ is

\frac{2}{3}

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3

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4

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\frac{3}{4}

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Solution

The correct option is B $3$
Given $γ = \frac{C_{P}}{C_{V}} = \frac{4}{3}$
and $d U = n (V d P + P d V)$
In a simple adiabatic process,
$Δ Q = Δ U + Δ W$
where $Δ Q = 0 = Δ U + Δ W o r = Δ U + P d V . . . (1)$

In a adiabatic process: $P V^{γ} = C P V^{4 / 3} = C$

Taking $l o g$ on both sides,
$l n P + \frac{4}{3} l n V = C 3 l n P + 4 l n V = C$

Differentiating the above equation
$3 \frac{d P}{P} + 4 \frac{d V}{V} = 0$
$3 V d P + 4 P d V = 0 3 V d P + 3 P d V + P d V = 0 . . . (2)$

On comparing $1$ and $2$ ,
$d U = 3 (V d P + P d V)$
Hence, $n = 3$

So, option B is correct.

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