Answer and explanation:
Given : Consider the Ideal Gas Law, [tex]PV=kT[/tex] where k>0 is a constant.
To find : Solve this equation for V in terms of P and T.
Solution :
[tex]PV=kT[/tex]
Divide each side by P,
[tex]V=\frac{kT}{P}[/tex] ....(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,
[tex]\frac{dV}{dP}=kT\frac{d}{dP}(\frac{1}{P})[/tex]
[tex]\frac{dV}{dP}=kT(-\frac{1}{P^2})[/tex]
[tex]\frac{dV}{dP}=-\frac{kT}{P^2}[/tex]
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,
[tex]\frac{dV}{dT}=\frac{k}{P}\frac{d}{dT}(T)[/tex]
[tex]\frac{dV}{dP}=\frac{k}{P}(1)[/tex]
[tex]\frac{dV}{dP}=\frac{k}{P}[/tex]
c) Assuming k =1, draw several level curves of the volume function and interpret the results.
When k=1, [tex]PV=T[/tex]
[tex]\frac{dV}{dP}=-\frac{T}{P^2}[/tex] <0
[tex]\frac{dV}{dP}=\frac{1}{P}[/tex] >0
Refer the attached figure below.
