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.
The work done by the gas during the expansion is 
. Use your answer in (i) to explain whether thermal energy is transferred to or from the gas during the expansion.
