Limits of specific heat of a gas

Consider 'm' mass of a gas enclosed in a cylinder fitted with an air-tight and frictionless piston.

1. Suppose the gas is suddenly compressed. No heat is supplied to the gas  i.e., ΔQ=0. But the temperature of the gas rises due to compression. Therefore,  c=ΔQ/mΔT=0/mΔT=0   i.e. the specific heat of the gas is zero.


2. Now the gas is heated and allowed to expand such that the rise in temperature of the gas due to the heat supplied is equal to the fall in temperature of the gas due to the expansion of the gas itself. Then the net rise in temperature is zero.i.e., ΔT=0. 
   Therefore, c=ΔQ/mΔT=ΔQ/m*0=infinite.  i.e.  the specific heat of the gas is infinite.
   
   
   


3. Again, the gas is heated and allowed to expand at such a rate that the fall in temperature due to expansion is less than the rise in temperature due to heat supplied. The temperature of the gas will rise i.e., ΔT is positive.
   Therefore,  c=ΔQ/mΔT=a positive value.  i.e., the specific heat of the gas is positive.
   
   
   


4. Finally, the gas is heated and allowed to expand at such a rate that the fall in temperature due to expansion is more than the rise in temperature due to heat supplied. The temperature of the gas will decrease i.e., ΔT is negative.
   Therefore, c=ΔQ/mΔT=a negative value. i.e., the specific heat of the gas is negative.

Thus the specific heat of gas may have any positive of negative value ranging form zero to infinity. The exact value depends on the conditions of pressure and volume when the gas is being heated. 

Note. Draw diagram of cylinder with piston