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# Internal Energy of an Ideal Gas

In this article we will discuss Internal Energy of an Ideal Gas

In this article, we will discuss Internal Energy of an Ideal Gas. So, let’s get started.

Internal Energy of an Ideal Gas

From the kinetic theory model we can obtain an expression for the total internal energy U of an ideal gas. The simplest form would be the sum of the kinetic energies of the molecules. If the gas contains N molecules:

U = Nmv²/2 (1)

Using mv² = 3kT, U can be expressed in terms of N and T:

U = 3/2NkT = 3/2nRT (2)

This is result is due to our definition of T and the properties of an ideal gas. If the number of molecules are fixed, the total energy U only depends on the temperature T. Another way of writing the equation is to use P and V. Since N kT = PV , we have U = (3/2)P V. However, it is nicer to recognize that U depends on only one parameter T.

Heat Capacity of an Ideal Gas at Constant Volume: Heat capacity relates the temperature increase to the energy transfered to the substance. For gases it is convenient to consider the heat capacity per mole of the gas. If the volume is held fixed during the process, no work is done by the gas, so Q = ∆U, so we have

Q = ∆U =3/2nR∆T (3)

So, cv = (3/2)R = 12.5 Joules/◦C per mole of the gas. The measured values of cv for the nobel gases (e.g. He, Ne, A, and Kr) are very close to this value for all temperatures where the elements are gases. The measured values of cv per mole for other gases are also close to 12.5 Joules/°C for low temperatures.