Internal Energy And Temperature

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Two identical samples of the same ideal gas are kept at the same temperature but stored at very different pressures in separate containers, so how do their internal energies per mole compare?

RankGuru · Accepted Answer

As clarified in NCERT Class 11, Chapter 12 (Thermodynamics), supported by kinetic theory, the internal energy of an ideal gas is a function of temperature alone, given per mole by $U = \dfrac{f}{2}RT$, where $f$ is the number of degrees of freedom. Since both samples are the same gas (same $f$) at the same temperature, their internal energies per mole are identical, regardless of the very different pressures or volumes. Pressure and volume can differ widely while $T$ stays fixed, as Boyle's law allows, but $U$ tracks only $T$. The option that the high-pressure sample has greater internal energy is wrong because pressure does not appear in the ideal-gas internal energy expression. The option favouring the low-pressure sample is wrong for the same reason. The option that internal energy depends mainly on container volume is wrong because, for an ideal gas, volume affects pressure but not internal energy at fixed temperature. A consistency check: setting equal $T$ and equal $f$ in $U = 	frac{f}{2}RT$ yields equal $U$ per mole, so the conclusion is internally consistent and independent of pressure.

Internal Energy And Temperature

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