# 2 moles of ideal gas initial pressure 202.6kpa,volume 4m^3compressed isothermally to final pressure of 303.9kpa. Find work done on gas.

hnystrom | Certified Educator

It is not clear from your question whether you want an approximate answer based on algebra or an exact answer using calculus.  I will give you both.

Since the process is isothermal this means that it is carried out at a constant temperature.  To do this it must be connected to a source of heat which provides the energy to do the work of compessing the gas.  If we assume an ideal gas then PV=nRT tells us that the product PV is a constant for each step of the way.  If we look at a graph of pressure versus volume we get a simple curve that connects the two points.  The work done is the area under that curve from the initial volume and pressure to the final volume and pressure.

If we don't know calculus we can approximate the area.  It looks like a trapezoid on its side.  The area of a trapezoid is the average of the two bases times the height.  The two bases are the two pressures and the height is the difference in volumes, so

W=1/2(Pfinal+Pinitial)(Vfinal-Vinitial)

If we know calculus we need the integral of the PdV from the initial volume to the final volume.  Since PV is a constant (isothem) the equation for P at any point is PinitialVinitial/V.  Taking the integral gives us

W=PinitialVinitial ln(Vfinal/Vinitial)

In both cases we need to know the final volume.

Vfinal=PinitialVinitial/Pfinal=2.67 cu. m

The apprximate answer = -336.8225 J

The exact answer is = -327.58 J.  Our approximation isn't too far off here.

The work is negative because we have to extract heat in order to reduce the volume and increase the pressure.  Extracting heat involves negative work because we are removing energy from the system.