Extension of the Reversible Adiabatic Equation to a General Equation for Ideal Gas

The reversible adiabatic equation for ideal gas (T $${\text{V}}_{\text{m}}^{\gamma{-1}}$$ = C) is one of the most important equations in thermodynamics. But there is still a lack of theoretical explanation of the constant C at present. Since the reversible adiabatic process is isentropic process, the constant C could be expressed as a function of entropy through the statistical thermodynamic method. According to the different heat capacities and heat capacity ratios, the molecules of ideal gas are divided into three sorts that are monatomic molecules, linear polyatomic molecules (which include the diatomic molecules) and nonlinear polyatomic molecules. Although the calculation formulas for entropy of different kinds of molecules are very different, their expressions of constant C are similar. All of them could be expressed as C = N × exp(Sm/CV,m), where N is structure parameter. The reversible adiabatic equation was extended to a more general equation T $${\text{V}}_{\text{m}}^{\gamma-1}$$ = N × exp(Sm/CV,m), which could be applied to any process for ideal gas.