Internal Energy U

Internal energy is the total energies inside a substance and is equal to the sum of

  • kinetic energies,
  • potential energies,
  • chemical energies and
  • nuclear energies of the molecules.

The kinetic energies of molecules include:

  • the translational energy,
  • the rotational kinetic energy and
  • the vibrational energy.

For gases, we usually only take the translational energy and the rotational kinetic energy into consideration, because only at higher temperature the vibrational energy plays then an important role. The average velocity is direct proportional to the temperature of gas. Therefore, temperature reflects the kinetic energies of molecules inside a substance and the internal energy as well. So we can say, higher temperature means a higher internal energy of a system.

The internal energy is additionally associated with potential energies, because there are interactions, binding forces, between the molecules.

The energy caused by atomic bonds is referred to as chemical energy. During a combustion process for example, some bonds are formed while others are destroyed. So the change of chemical energy occurs.

U stands for internal energy. Its unit is J.

u stands for specific internal energy per unit mass of a system. Its unit is J/kg.

As mentioned above, the kinetic energies of molecules of gas depend on temperature T and the latent energy depends on specific volume of gas ν. Therefore, specific internal energy is a function depending on temperature T and specific volume ν:

u=f (T,ν)

Moreover, pressure p is related to temperature T and specific volume ν. So internal energy can also be written in:

u=f (p,ν)     or    u=f (T,p)

Because the internal energy is a property of state, its change can be described in form of total differential:

u=f (T,ν):

    Unbenannte QQ-Bildschirmaufnahme20130304165103

u=f (p,ν):

Unbenannte QQ-Bildschirmaufnahme20130304165103

u=f (T,p):


If gas is consideres as ideal gas, then there are no interactions between molecules which means no latent energy. Hence, internal energy is only dependent on temperature T:

u=f (T)

(available only for ideal gas!)

Since internal energy is point function and therefore path-independent, so:


cyclic int u

And its cyclic integral is:

inte u