## Physics Friday 74

Previously, I discussed the heat capacity of an ideal gas, and the relationship to the molecular degrees of freedom f. Now, another important property for a gas is the ratio of the specific heat for constant pressure to that for constant volume, known as the heat capacity ratio, or adiabatic index:
. Recalling the derivation of the specific heats from the internal energy and enthalpy, we see that for an ideal gas, we can also express the adiabatic index as .

Recalling that in terms of f, our ideal gas has heat capacities  and , so . Thus, a monatomic ideal gas has f=3, and so , while a diatomic gas with five degrees of freedom (such as nitrogen or oxygen at room temperature) has . For comparison, room temperature air has an adiabatic index measured at approximately 1.403, very close to the 7/5=1.4 given above. Note that as f increases, as occurs for real gases with increasing temperature, γ decreases toward unity.

To see why γ is called the adiabatic index, we recall that an adiabatic process is one in which no heat is transfered to or from the working fluid. We recall that the increment in internal energy dU is the sum of the heat added  and the work done ; as the heat transfered is zero (, dS=0), we have
. Now, as , we get

Now, putting the ideal gas law  in differential form, . Solving for dT and using the difference relation , we see

Equating with the above condition for dT in an adiabatic process:
.
Integrating the last, we obtain
, which we can combine using the properties of logarithms to get:
 for an adiabatic process; thus the use of the term adiabatic index. The above condition for pressure and volume is known as the adiabatic condition.