For an adiabatic process involving an ideal gas (A) \(\mathrm{P}^{\gamma-1}=\mathrm{T}^{\gamma-1}=\) constant (B) \(\mathrm{P}^{1-\gamma}=\mathrm{T}^{\gamma}=\) constant (C) \(\mathrm{PT}^{\gamma-1}=\) constant (D) \(\mathrm{P}^{\gamma-1}=\mathrm{T}^{\gamma}=\) constant

An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings. For an ideal gas, the relationship between pressure, volume, and temperature is given by the ideal gas law: \[PV=nRT\] where: P = pressure V = volume T = temperature n = moles of the gas, and R = universal gas constant In an adiabatic process, the ratio of the specific heats γ (gamma) is important. It is given by: \[\gamma = \frac{C_p}{C_v}\] where: \(C_p\) = molar specific heat at constant pressure \(C_v\) = molar specific heat at constant volume

For an adiabatic process involving an ideal gas, the relationship between pressure, volume, and temperature can be derived using the equations mentioned above. Let's first derive the relationship between the pressure and the volume in an adiabatic process. Consider the following equation for an ideal gas: \[PV^{\gamma}=K_1\] where K₁ is a constant. Now, let's derive an equation for the relationship between pressure and temperature. Using the ideal gas law, \[PV=nRT\] \[T=\frac{PV}{nR}\] Substitute the value of T in the adiabatic equation above: \[P(\frac{PV}{nR})^{\gamma-1}=K_2\] Simplifying, we obtain the relationship between pressure and temperature in an adiabatic process: \[P^{\gamma}T^{\gamma-1} = K_3\] or \[P^{\gamma-1} = \frac{K_3}{T^{\gamma-1}} = \text{constant}\]

Now that we have derived the equation relating pressure and temperature in an adiabatic process: \[P^{\gamma-1} = \text{constant} * T^{\gamma-1}\] Comparing this with the given options: (A) \(\mathrm{P}^{\gamma-1}=\mathrm{T}^{\gamma-1}=\) constant (B) \(\mathrm{P}^{1-\gamma}=\mathrm{T}^{\gamma}=\) constant (C) \(\mathrm{PT}^{\gamma-1}=\) constant (D) \(\mathrm{P}^{\gamma-1}=\mathrm{T}^{\gamma}=\) constant It can be seen that option (A) matches the derived equation, so it is the correct answer. Thus, for an adiabatic process involving an ideal gas, the relation between pressure and temperature is: \(\mathrm{P}^{\gamma-1}=\mathrm{T}^{\gamma-1}=\) constant