# Partial Derivative Manipulation for Physical Chemistry Homework problem

1. Sep 24, 2014

### Jayjayjay

1. The problem statement, all variables and given/known data
Given the functions Q(v,w) and R(v,w)

K = v(dQ/dv)r and L = v(dQ/dv)w

Show that
(1/v)K = (1/v)L + [B](dQ/dw)v (dW/dv)r [/B]
[B]
I have the problem attached if for clarity of the information.[/B]

2. Relevant equations
I assume everything is given in the problem.

3. The attempt at a solution
Q(v,w) --> dQ = (dQ/dv)w DV + (dQ/dw)v DW
R(v,w) --> dR = (dR/dv)w DV + (dR/dw)v DW

(1/v)K = (dQ/dv)r (Equation 1)
so I have to manipulate the given information to equal (dQ/dv)r and substitute (I think)

(1/v)L = (dQ/dv)w
which is in the function Q(v,w) so I was thinking you could just make

dQ - (dQ/dw)v DW = (dQ/dv)w

and substitute to equal

(1/v)L = dQ - (dQ/dw)v dW --> (1/v)L + (dQ/dw)v dW = dQ

If I divided the whole thing by dv I could get

(1/v)L + (dQ/dw)v (dW/dv)r = (dQ/dv)r

and substitute this into equation 1 to get

(1/v)K = (1/v)L + [B](dQ/dw)v (dW/dv)r [/B]
[B]
Is this correct? I'm not sure. [/B]

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2. Sep 24, 2014

### Staff: Mentor

If R=R(v,w), what is dR in terms of dv and dw (and the partial derivatives of R with respect to v and w)? How do dv and dw have to vary if dR = 0? What is the partial derivative of w with respect to v at constant R?

Chet