An ideal gas is expanding such that PT^{3} = | vedclass

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(C) Given the process equation: $PT^{3} = 	ext{constant}$. Using the ideal gas law $PV = nRT$, we can write $P = nRT/V$. Substituting $P$ into the gi

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$4$

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(C) Given the process equation: $PT^{3} = 	ext{constant}$. Using the ideal gas law $PV = nRT$, we can write $P = nRT/V$. Substituting $P$ into the given equation: $(nRT/V) 	imes T^{3} = 	ext{constant}$. Since $n$ and $R$ are constants, this simplifies to $T^{4}/V = 	ext{constant}$, or $T^{4} = kV$ where $k$ is a constant. Differentiating both sides, we get $4T^{3} dT = k dV$. Since $k = T^{4}/V$, we substitute $k$: $4T^{3} dT = (T^{4}/V) dV$. Dividing both sides by $T^{3}$, we get $4 dT = (T/V) dV$, which rearranges to $dV/V = 4(dT/T)$. The coefficient of volume expansion $\gamma$ is defined by the relation $dV = V \gamma dT$, or $\gamma = (1/V) (dV/dT)$. From our derived relation $dV/V = 4(dT/T)$, we have $(1/V) (dV/dT) = 4/T$. Therefore, the coefficient of volume expansion is $\gamma = 4/T$.

An ideal gas is expanding such that $PT^{3} = \text{constant}$. The coefficient of volume expansion of the gas is: (in $/T$)

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