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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.

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    $\begingroup$ As Cv=x.R and Cp=Cv+R, I do not think it is possible. $\endgroup$ – Poutnik May 2 at 17:01
  • $\begingroup$ But putting the values of monotomic gas and n as 4/3 (both being valid and within domain) seem to give a negative value. $\endgroup$ – evamPUNdit May 2 at 17:06
  • $\begingroup$ 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. $\endgroup$ – Poutnik May 2 at 17:07
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    $\begingroup$ Monoatomic gas has Cv=3/2R, Cp=5/2R, gamma 5/3. $\endgroup$ – Poutnik May 2 at 17:10
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    $\begingroup$ 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. $\endgroup$ – Chet Miller May 2 at 19:31
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I'm going to continue to resist calling C the heat capacity because, as used here, it is not path-independent. Instead, I will take it to merely represent the derivative of the heat added per mole of ideal gas with respect to temperature for a polytropic process path.

With this caveat, here is my interpretation of what is happening mathematically when C takes on negative values: If the gas is expanding, a negative value of C means that we added heat to the gas during the expansion, but not enough to offset the expansion cooling that would occur in an adiabatic reversible expansion, so its temperature decreases while we add heat. If the gas is being compressed, a negative value of C means that we removed heat from the gas during the compression, but not enough to offset the compression heating that would occur in an adiabatic reversible compression, so its temperature increases while we remove heat. It's as simple as that.

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