Which of the following is NOT a correct equation for adiabatic...

For an ideal gas undergoing a reversible adiabatic process, the fundamental relationship between pressure ($P$) and volume ($V$) is given by:

$PV^{\gamma} = \text{constant}$

Using the ideal gas equation, $PV = nRT$, we can express volume as $V = \dfrac{nRT}{P}$. Substituting this into the adiabatic equation:

$P \left( \dfrac{nRT}{P} \right)^{\gamma} = \text{constant}$

$P \cdot P^{-\gamma} \cdot T^{\gamma} = \dfrac{\text{constant}}{(nR)^{\gamma}}$

$P^{1-\gamma}T^{\gamma} = \text{constant}$

This confirms that $P^{1-\gamma}T^{\gamma} = \text{constant}$ is a valid equation for an adiabatic process, which corresponds to option (3).

Similarly, we can substitute $P = \dfrac{nRT}{V}$ into the original equation:

\left( \dfrac{nRT}{V} \right) V^{\gamma} = \text{constant}$<br><br>$T V^{\gamma-1} = \dfrac{\text{constant}}{nR} = \text{constant}$<br><br>This confirms that$TV^{\gamma-1} = \text{constant}$is also valid, which corresponds to option (2).<br><br>Option (4) states$P^{\gamma-1}T^{\gamma-1} = \text{constant}$, which contradicts the derived relation$P^{1-\gamma}T^{\gamma} = \text{constant}$. Therefore, option (4) is NOT a correct equation for an adiabatic process.<br><br>Answer:$P^{\gamma-1}T^{\gamma-1} = constant