Internal energy
Internal energy of the ideal gas

Thermodynamics often uses the concept of the ideal gas for teaching purposes, and as an approximation for working systems. The ideal gas consists of particles considered as point objects that interact only by elastic collisions and fill a volume such that their mean free path between collisions is much larger than their diameter. Such systems approximate monatomic gases such as helium and other noble gases. For an ideal gas the kinetic energy consists only of the translational energy of the individual atoms. Monatomic particles do not possess rotational or vibrational degrees of freedom, and are not electronically excited to higher energies except at very high temperatures.

Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): U = U (N, T) {\displaystyle U=U(N,T)}. It is not dependent on other thermodynamic quantities such as pressure or density.

The internal energy of an ideal gas is proportional to its amount of substance (number of moles) N {\displaystyle N} and to its temperature T {\displaystyle T}

where c V {\displaystyle c_{V}} is the isochoric (at constant volume) molar heat capacity of the gas; c V {\displaystyle c_{V}} is constant for an ideal gas. The internal energy of any gas (ideal or not) may be written as a function of the three extensive properties S {\displaystyle S}, V {\displaystyle V}, N {\displaystyle N} (entropy, volume, number of moles). In case of the ideal gas it is in the following way

where c o n s t {\displaystyle \mathrm {const} } is an arbitrary positive constant and where R {\displaystyle R} is the universal gas constant. It is easily seen that U {\displaystyle U} is a linearly homogeneous function of the three variables (that is, it is extensive in these variables), and that it is weakly convex. Knowing temperature and pressure to be the derivatives T = ∂ U ∂ S, {\displaystyle T={\frac {\partial U}{\partial S}},} P = − ∂ U ∂ V, {\displaystyle P=-{\frac {\partial U}{\partial V}},} the ideal gas law P V = N R T {\displaystyle PV=NRT} immediately follows as below:
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