$)$ Explain how a Bourdon Gauge can be used to measure the pressure in a container of gas.

$[6$ marks$]$

$($b$)$ The total average pressure P$,$ exerted on a wall of a container of volume V is: where m is the mass of an individual molecule, N is the total number of molecules in the vessel and vi is the velocity of molecule i$.$ Use this relationship to develop a formula for the root mean square $($rms$)$ velocity for the system, represented by c$.$

$[8$ marks$]$

$($c$)$ Using your results from $($b$),$ and the ideal gas law PV $=$ N kb T$,$ show that the mechanical kinetic energy of a molecule is E $=3/2$ kb T$.$