# Negative Molar Specific Heat of a Gas

It is known that the specific heat of a gas is process dependent. So it must be theoretically possible to have a negative value for a gas according to the following equation (for polytropic process, $$PV^n = \mathrm{const}$$):

$$C = \frac R{γ - 1} + \frac R{1 - n}$$

where $$C$$ is molar specific heat and $$γ$$ is adiabatic exponent.

Supposing $$γ$$ is $$\frac{5}{3}$$ and $$n$$ is $$\frac{4}{3}$$, $$C$$ comes out to be negative. Is it practically possible and if so what would it signify? As you provide more heat to a gas in such a process, would it lose temperature? Please clarify.

• As Cv=x.R and Cp=Cv+R, I do not think it is possible. May 2, 2019 at 17:01
• But putting the values of monotomic gas and n as 4/3 (both being valid and within domain) seem to give a negative value. May 2, 2019 at 17:06
• While it is possible to add heat to gas and get the final temperature lower, one has to count done work and also possible change of C. May 2, 2019 at 17:07
• Monoatomic gas has Cv=3/2R, Cp=5/2R, gamma 5/3. May 2, 2019 at 17:10
• It is a very bad idea to consider to consider heat capacity (a function of state) to be a function of process path (such as dQ/dT). Doing this will drive you nuts in the future. In thermodynamics, we purposely define heat capacity as a function of either enthalpy or internal energy, both of with are functions of state (and not functions of path). Problems like your polytropic gas example force you to think of heat capacity in the wrong (and non-productive) way. May 2, 2019 at 19:31