We present an alternative derivation of the relation between temperature and volume for a reversible, adiabatic process involving an ideal gas. The derivation for a monatomic gas starts with the Sackur–Tetrode equation and takes only one step. We also address the extension to diatomic gases.

*P*, volume

*V*, and temperature

*T*) in an adiabatic process involving an ideal gas are essential for the analysis of thermodynamic engines, including the Carnot, Diesel, Otto, and Rankine cycles as well as real refrigerator cycles, all of which include adiabatic compression or expansion strokes. Standard expressions for such relations can be derived starting with the first law of thermodynamics for an adiabatic process (where the energy transferred as heat is

*Q = 0*), which becomes

*dU = đW*, or

*f*is the number of active quadratic degrees of freedom in the gas,

*N*is the number of gas molecules in the system, and

*k*is the Boltzmann constant. Replacing

*P = NkT/V*and separating variables yield

*VT*

^{f/}^{2}= constant. This calculation is presented in most thermodynamics textbooks (see Ref. 1, for example). Using the ideal gas law to eliminate

*T*gives the expression

*PV*= constant, where

^{γ}*γ = C*(

_{P}/C_{V}.*C*and

_{P}*C*are, respectively, the heat capacities at constant pressure and volume, and

_{V}*γ*is often called the adiabatic constant). This last form is particularly useful for cycle analysis in

*PV*diagrams. We refer to both the

*V-T*and

*P-V*relations as the adiabatic equations for the ideal gas.

*VT*

^{f/}^{2}= constant for the special case of a classical monatomic ideal gas using (1) the fact that adiabatic, reversible processes are isentropic (adiabatic, irreversible processes, which include free expansions, are outside the scope of this Note), and (2) the Sackur–Tetrode equation,

^{2–7}a well-known result in statistical physics. This equation, originally derived by Otto Sackur and Hugo Tetrode in 1911–1913, expresses the entropy of a monatomic ideal gas in terms of several combinations of variables. Here, we use the form

*Λ*is the thermal de Broglie wavelength

*VT*

^{3/2}must also remain constant. This is nothing other than the

*V-T*adiabatic equation for three degrees of freedom—those of the monatomic ideal gas! Another version of the Sackur–Tetrode equation is

*U*is proportional to

*T*.

The above derivation does not generate a new result. Instead, it confirms the adiabatic equation through a very different route from the one that students usually learn. The route here comes from statistical mechanics, while the usual one is strictly thermodynamic. Such alternative derivations strengthen the connections within, and the consistency of thermal physics, for the benefit of the students. Despite the disparity of both methods, we remark that the power of 3 in the Sackur–Tetrode equation is directly associated with the three (translational) degrees of freedom of the monatomic ideal gas.

The second, and final purpose of this Note is to extend the result to the more common diatomic gases. In this case, internal types of motion need to be taken into account. The monatomic partition function is multiplied by that of the internal degrees of freedom, *Z _{int}*. In the high-temperature/classical limit, for each internal degree of freedom,

*Z*, where

_{int}∼ kT/E*E*is some characteristic energy spacing (

*ℏ*

^{2}

*/2I*for the rotation of a molecule of moment of inertia

*I*and

*ℏω*for vibration of frequency

*ω*). Thus, the argument of the logarithm in Eq. (3) will be multiplied by a factor

*kT/E*for each degree of freedom, as shown in Ref. 1, p. 255. This bumps up the argument of the logarithm to

*VT*

^{5/2}and

*VT*

^{7/2}, consistent with

*f = 5*and

*f = 7*, for diatomic gases, respectively, as these types of internal motion become active. This result agrees with the expressions derivable by classical thermodynamics.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.