Show that the thermodynamic work done in compressing a gas at a constant pressure is given by the product of that pressure and volume?

To show that the work done in compressing a gas at a constant pressure is given by the product of that pressure and volume, we can use the first law of thermodynamics, which states:

ΔU = Q - W

Where:
ΔU = change in internal energy of the gas
Q = heat added to the gas
W = work done on the gas

Since the process is done at constant pressure, we can write the work done as:

W = -PΔV

Where:
P = pressure
ΔV = change in volume

Since the pressure is constant, we can rewrite the work done as:

W = -P(Vf - Vi)

Where:
Vf = final volume
Vi = initial volume

Substitute this expression for work done into the first law of thermodynamics:

ΔU = Q + P(Vf - Vi)

Since the process is adiabatic (no heat exchange), Q = 0:

ΔU = P(Vf - Vi)

Since the change in internal energy is given by the ideal gas law as:

ΔU = nCvΔT

Where:
n = number of moles of gas
Cv = molar specific heat at constant volume
ΔT = change in temperature

We can equate the two expressions for ΔU:

nCvΔT = P(Vf - Vi)

Rearranging the equation, we get:

PΔV = nCvΔT

Since the ideal gas law states:

PV = nRT

We can rewrite the equation as:

PΔV = nRΔT

Therefore, the work done in compressing a gas at a constant pressure is given by the product of that pressure and volume.