In an ideal gas we can consider the internal energy to be due to the kinetic energy of the particles.
There are 3 degrees of freedom for a monotomic gas: they can move in the x, y, and z directions.
Each degree of freedom contributes #1/2kT# of energy which makes#3/2kT# in total.
So total internal energy #U# is given by:
#U=3/2kT#
#k# is the gas constant per mole and is The Boltzmann Constant.
#k=1.38xx10^(-23)m^(2).kg.s^(-2).K^(-1)#
#T# is the absolute temperature.
A diatomic gas like nitrogen #N_2# has 5 degrees of freedom. 3 for translational movement (x,y and z) and 2 for rotation.
This makes 5 in all. So for a diatomic gas:
#U=5/2kT#
In our example we have 4.5 moles of gas so:
#U_("tot")=5/2xxLxx4.5xxkT#
#L# is the Avogadro Constant = #6.02xx10^(23)mol^(-1)#
#U_("tot")=5/2xx6.02xxcancel(10^(23))xx4.5xx1.38xxcancel(10^(-23))xx526#
#U_("tot")=49160.2"J"#
#=49.16"kJ"#
