# Ideal gas equation for adiabatic process

In an adiabatic expansion or compression of gases, does the ideal gas equation change from $$PV=nRT$$ to $$PV^{\gamma} = nRT$$? If so, then why as pressure and volume are inversely related in both the cases and the only difference is in their heat exchange process with the surroundings? Why is only $$\gamma = C_{p}/C_{v}$$ used in the equation as neither volume nor pressure is constant here?

In an adiabatic expansion or compression of gases, does the ideal gas equation change from 𝑃𝑉=𝑛𝑅𝑇 to $$𝑃𝑉^𝛾=𝑛𝑅𝑇$$

No. See the equation PV = nRT is valid for an ideal gas for any process.

In Adiabatic process, $$PV^𝛾 = k ≠ nRT$$where k is a constant.

However Ideal gas equation equally holds true. $$PV = nRT$$ here P,V and T are variable.

Why is only 𝛾=𝐶𝑝/𝐶𝑣 used in the equation as neither volume nor pressure is constant here?

See in the cases of ideal gases, Cp and Cv are treated as constants.Even though P and V are not constant, heat capacity at constant Pressure and constant Volume - Cp and Cv are constants for ideal gas.And their ratio is called Adiabatic Index or Gamma.

P.S. If you still have any doubt,comment below.

• In adiabatic process P.V is not constant as temperature is not the same but PV/T =nR =constant PV^gamma =k. Is V/T is same as V^gamma – user270071 Aug 26 at 4:57
• @user270071 Just because two quantities are constant, it doesn't mean that they are the same! The two quantities that you're comparing don't even have the same dimension... – Philip Aug 26 at 5:35
• @user270071 I agree with Philip on this one. – Tony Stark Aug 26 at 5:37