Question

Derive the law of Thermodynamics that W= P Delta V.

Answer 1

Dear Student,

Now,imagine a system that consists of a sample of ammonia trapped in a piston and cylinder as shown in figure.
Assume that the pressure of the gas pushing up on the piston just balances the weight of the pistion, so that the volume of the gas is constant,
Now assume that the gas decomposes to form nitrogen and hydrogen, increasing the number of gas particles in the container,
If the temperature and pressure of the gas are held constant, this means volume of the gas must increase.
$2{\mathrm{NH}}_3\left(\mathrm{g}\right)\to {\mathrm{N}}_2\left(\mathrm{g}\right)\hspace{0.17em}+3{\mathrm{H}}_2\left(\mathrm{g}\right)$

The volume of the gas can increase by pushing the piston partway out of the cylinder. The amount of work done is equal to the product of the force exerted on the piston times the distance the piston is moved.
w= F x d

The pressure (P) the gas exerts on the piston is equal to,
$\mathrm{P}=\frac{\mathrm{F}}{\mathrm{A}}$
or, F= P x A

or, w =(P x A) x d
The product of the area of the piston times the distance the piston moves is equal to the change that occurs in the volume of the systame when the gas expands,
So, ΔV= A x d
Therefore, the magnitude of work done will be
w= PΔV