Question

Consider the Ideal Gas Law

Answer 1

Answer and explanation:

Given : Consider the Ideal Gas Law, $PV=kT$ where k>0 is a constant.

To find : Solve this equation for V in terms of P and T.

Solution :

$PV=kT$

Divide each side by P,

$V=\frac{kT}{P}$ ....(1)

a) Determine the rate of change of the volume with respect to the pressure at constant temperature. Interpret the result.

Differentiate equation (1) w.r.t P,

$\frac{dV}{dP}=kT\frac{d}{dP}(\frac{1}{P})$

$\frac{dV}{dP}=kT(-\frac{1}{P^2})$

$\frac{dV}{dP}=-\frac{kT}{P^2}$

​b) Determine the rate of change of the volume with respect to the temperature at constant pressure. Interpret the result.

Differentiate equation (1) w.r.t T,

$\frac{dV}{dT}=\frac{k}{P}\frac{d}{dT}(T)$

$\frac{dV}{dP}=\frac{k}{P}(1)$

$\frac{dV}{dP}=\frac{k}{P}$

c) Assuming k =1,  draw several level curves of the volume function and interpret the results.

When k=1, $PV=T$

$\frac{dV}{dP}=-\frac{T}{P^2}$ <0

$\frac{dV}{dP}=\frac{1}{P}$ >0

Refer the attached figure below.