An ideal gas is contained in a cylinder by means of a movable frictionless piston, as illustrated in the figure below.
Initially, the gas has a volume of 
at a pressure of 
and a temperature of 
.
Show that the number of gas molecules in the cylinder is 
.
The gas expands so that its volume increases to 
at a pressure of 
and a temperature of 
, as shown in the figure below.
The average translational kinetic energy 
of a molecule of an ideal gas is given by
where 
is the Boltzmann constant and 
is the thermodynamic temperature. Calculate the increase in internal energy 
of the gas during the expansion.
