Question

For adiabatic Process which relation is true mentioned below ? \(\gamma=\left\\{\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right\\}\) (A) \(\mathrm{p}^{\gamma} \mathrm{V}=\mathrm{Const}\) (B) \(\mathrm{T}^{\gamma} \mathrm{V}=\mathrm{Const}\) (C) TV \(^{\gamma}=\) Const (D) \(\mathrm{TV}^{\gamma-1}=\) Const

Short answer

The correct relation for the adiabatic process is (D) \(TV^{γ-1} = Const\).

Step-by-step solution

Recognize the adiabatic process

In an adiabatic process, there is no heat transfer into or out of the system, i.e., the system is thermally isolated. The equation governing adiabatic processes for an ideal gas can be obtained by combining the ideal gas law and the definition of adiabatic processes.

Recall the ideal gas law

The ideal gas law is given by the equation: \(pV=nRT\), where p is pressure, V is volume, n is the number of moles of the gas, R is the gas constant, and T is temperature.

Derive the adiabatic process equation

For an adiabatic process, the relation between pressure, volume, and temperature can be derived from the ideal gas law and the adiabatic index (γ). By rearranging the ideal gas law and using the adiabatic index, we get the adiabatic process equation: \(pV^γ = C\), where C is a constant.

Compare the given options

Now let's compare the given options with the adiabatic process equation we derived: (A) \(p^γV = Const\) - This is not the correct relation. (B) \(T^γV = Const\) - This is not the correct relation. (C) \(TV^γ = Const\) - This is not the correct relation. (D) \(TV^{γ-1} = Const\) - This is the correct relation. By comparing the given options with the derived adiabatic process equation, we find that the correct relation for the adiabatic process is (D) \(TV^{γ-1} = Const\).