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For an ideal gas in this special case, the internal energy, U, is only a function of the temperature T; therefore the partial derivative of heat capacity with respect to T is identically the same as the full derivative, yielding through some manipulation
Further manipulation using the differential version of the ideal gas law, the previous equation, and assuming constant pressure, one finds
For an adiabatic process and recalling , one finds
One can solve this simple differential equation to find
This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows
where is the mean molecular weight of the gas or plasma; and is the mass of the Hydrogen atom, which is extremely close to the mass of the proton, , the quantity more often used in astrophysical theory of galaxy clusters. This is what astrophysicists refer to as "entropy" and has units of [keV cm2]. This quantity relates to the thermodynamic entropy as
where , the density of states in statistical theory, takes on the value of K as defined above.