Question

Consider the Ideal Gas Law

PV = ​kT,

where k>0

is a constant. Solve this equation for V in terms of P and T.

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

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

​c) Assuming k =1,

draw several level curves of the volume function and interpret the results.

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=(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,

`(dV)/(dP)=kT(d)/(dP)((1)/(P))`

`(dV)/(dP)=kT(-(1)/(P^2))`

`(dV)/(dP)=-(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,

`(dV)/(dT)=(k)/(P)(d)/(dT)(T)`

`(dV)/(dP)=(k)/(P)(1)`

`(dV)/(dP)=(k)/(P)`

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

When k=1,
`PV=T`

`(dV)/(dP)=-(T)/(P^2)` <0

`(dV)/(dP)=(1)/(P)` >0

Refer the attached figure below.