P)_r = 1, proving the desired result.

[LoopQG]

Homework Statement

Given the Equation of State f(P,V,T)=0 and r=r(P,V,T) be a constraint show that

these are all partials i can't get latek to work for me sorry about that

(dp/dV)_r (dV/dT)_r (dT/dP)_r = 1

the _r means holding r constant.

hint consider dr

I have tried several different ways but I think I am starting incorrectly, any hint or tips are welcome.

my first thought is

take $$df= (\partial f/ \partial P)dp + (\partial f/\partial V)dV + (\partial f/\partial T)dT = 0$$

and similarly do dr but I don't know how to connect f and r.

Any help appreciated.

 

[NateD]

Thank youHomework Equations Equation of State f(P,V,T)=0 r=r(P,V,T) The Attempt at a Solution Take df= (\partial f/ \partial P)dp + (\partial f/\partial V)dV + (\partial f/\partial T)dT = 0 and dr= (\partial r/ \partial P)dp + (\partial r/\partial V)dV + (\partial r/\partial T)dT = 0 Since we are holding r constant dr = 0 Using this we can rearrange the equation to get dp/(\partial f/\partial P) + dV / (\partial f/\partial V) + dT / (\partial f/\partial T) = 0Now taking partial derivatives with respect to one variable at a time while holding all other variables constant (dp/dV)_r (dV/dT)_r (dT/dP)_r = 1We get (dp/dV)_r = -(\partial f/\partial V)/(\partial f/\partial P) (dV/dT)_r = -(\partial f/\partial T)/(\partial f/\partial V) (dT/dP)_r = -(\partial f/\partial P)/(\partial f/\partial T) Multiplying these three together will give us (dp/dV)_r (dV/dT)_r (dT/dP)_r = (\partial f/\partial V)(\partial f/\partial T)(\partial f/\partial P)/(\partial f/\partial P)(\partial f/\partial V)(\partial f/\partial T) which simplifies to (dp/dV)_r (dV/dT)_r (dT/d