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 
.
