Find Isobaric Expansion & Pressure-Volume Coefficient for Solid

[ChiralSuperfields]

Homework Statement

The problem I am trying to solve is,
Find the isobaric expansion coefficient ##\frac{dV}{dT}## and the isothermal pressure-volume coefficient ##\frac{dV}{dP}## of a solid that has equation of state ##V+bpT–cT^2=0##

Relevant Equations

Equation of state of solid: ##V+bpT–cT^2=0##

The answer to this problem is

However, I am confused how this relates to the question.

My working is,
##V = cT^2 - bpT##
##\frac{dV}{dT} = 2cT - bp## (I take the partial derivative of volume with respect to temperature to get the isobaric expansion coefficient)

##\frac{dV}{dP} = 0## (I take the partial derivative of volume with respect to pressure to get the isothermal pressure-volume coefficient)

If someone please knows whether I am correct or not then that would be greatly appreciated!

Many thanks!

 

[haruspex]

I suspect that p and P are supposed to be the same.

 

[ChiralSuperfields]

I suspect that p and P are supposed to be the same.

Thank you for your reply @haruspex!

That gives ##\frac{dV}{dP} = -bT##. Do you please know whether the textbook solution is wrong? It appears they just give a polynomial in temperature.

Many thanks!

 

[haruspex]

Thank you for your reply @haruspex!

That gives ##\frac{dV}{dP} = -bT##. Do you please know whether the textbook solution is wrong? It appears they just give a polynomial in temperature.

Many thanks!

It is obviously not an answer to the question. It merely restates the given equation. The answers must take the form ##\frac{dV}{dT}=## etc. Or better, ##\frac{\partial V}{\partial T}=## etc.
Perhaps the original question was the reverse: it provided the partial derivatives and asked for the state equation. Someone changed the question but forgot to change the answer.

 

[ChiralSuperfields]

It is obviously not an answer to the question. It merely restates the given equation. The answers must take the form ##\frac{dV}{dT}=## etc. Or better, ##\frac{\partial V}{\partial T}=## etc.
Perhaps the original question was the reverse: it provided the partial derivatives and asked for the state equation. Someone changed the question but forgot to change the answer.

Thank you for your help @haruspex ! I understand now :)