The internal energy is simply the total kinetic energy
of all molecules in the gas:
where I use <...> to denote an average over all the atoms of the gas.
can be related to the temperature of the gas through the ideal gas law as follows.
It can easily be shown that
The derivation is in the book an its conceptual content is that
the pressure exerted by a gas arises from incessant elastic collisions with the
atoms in the gas. The mean squared velocity appears in this expression because
the rate of impact carries a factor of velocity as does the momentum transfer in
each collision. The factor of one third comes about because we live in a three dimensional
space and therefore
Where 
is the mean squared average of the projection of the velocity
of individual atoms on a specific (arbitrary) spatial direction.
Combining Eq. 2
with the ideal gas law:
yields
It is important to remember that there is a wide distribution of velocities for
atoms in a gas. The average of the velocities vanishes since there is no preferred
direction for a gas in thermodynamic equilibrium. The average squared speed however
is given by the expression above. The full velocity distribution was derived
by J. C. Maxwell and is discussed to some extent in the book.
We can now eliminate the microscopic variable 
and obtain the desired expression
which relates the internal energy to the temperature of the gas:
Example We calculate energetics of a specific quantity of gas. Say we have
one mole of an ideal mon-atomic
gas at room temperature, what is the total internal energy:
This enough energy to keep a 60 W bulb on for about one minute or to lift
a mass of 1 kg to a height of 381 m. We can also easily derive the mean squared
velocity of atoms in the gas by turning Eq. 5 inside out:
Putting in numbers for helium gas where 
kg/mole
we get
We see that while the mean kinetic energy per molecule for a gas
at a specified temperature is independent of the type of gas, the
Root Mean Squared (RMS)
speed is inversely proportional to the square root of the mass
of the molecule.
Eq. 6 is important because it allows us to derive an expression for the
specific heat of an ideal mon-atomic gas. When we analyzed the isochoric process we defined
the constant volume specific heat as
and showed that we could write the internal energy as
By equating Eq. 6 and Eq. 11 we get
We can also write an expression for the constant pressure specific heat
of a mon-atomic gas:
Thus we have finally an expression for the constant
which has appeared in various contexts such as for the relationship
between pressure and volume during an adiabatic process:
and also in the expression for the velocity of sound in gas:
