An ideal gas has a volume of 
at a pressure of 
and a temperature of 
, as shown in the figure below.
The gas suddenly expands to a volume of 
. During the expansion, no thermal energy is transferred. The final pressure of the gas is 
at temperature 
, as shown in the figure above.
Show that the number of gas molecules is 
.
Show that the final temperature of the gas is 
.
The average translational kinetic energy 
of a molecule of an ideal gas is given by
where 
is the thermodynamic temperature and 
is the Boltzmann constant. Calculate the increase in internal energy 
of the gas.
Use the first law of thermodynamics to explain why the external work 
done on the gas during the expansion is equal to the increase in internal energy in (b)(ii).
