Internal energy of an ideal gas

We have seen that in an ideal gas the internal energy is entirely kinetic. Kinetic theory takes a further step and links it directly to temperature: the more the molecules agitate, the higher the temperature. The quantitative result is one of the chapter’s fundamental formulae.

Law — Internal energy of an ideal gas

$\ev{E_{\text{int}} = \frac{f}{2}\,N\,k_B\,T = \frac{f}{2}\,n\,R\,T}$ where $f$ is the number of degrees of freedom of the molecule.

The formula exists in two equivalent versions: one “per molecule” (with $N$ molecules and the Boltzmann constant $k_B$) and one “per mole” (with $n$ moles and the gas constant $R$). The bridge between the two is the relation $R = N_A\,k_B$, i.e. the gas constant is simply the Boltzmann constant multiplied by Avogadro’s number.

Key formula — Constants

$k_B = 1{,}38\cdot 10^{-23}\;\text{J/K}$ $R = N_A\,k_B = 8{,}31\;\text{J/(mol·K)}$ Monatomic: $f = 3$; diatomic: $f = 5$.

The physical intuition is powerful: temperature is not a substance or a fluid, but a measure of agitation. Doubling the absolute temperature of an ideal gas means doubling its internal energy, i.e. the total kinetic energy of its molecules. Absolute zero ($T = 0$ K) corresponds, in this classical model, to zero agitation: the molecules would be at rest.

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Topics: Teoria cinetica dei gas Concepts: Energia interna · Temperatura Skills: Interpretazione micro-macro

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