# Express volume expansivity in terms of density

1. Jan 22, 2017

### matthewsyq1995

1. The problem statement, all variables and given/known data
Express volume expansivity (B) in terms of density (ρ) and its partial derivatives
2. Relevant equations
B = (1/V) (dV/dT)
V = m/ρ

3. The attempt at a solution
I have only managed to substitute m/ρ into the expansivity equation.
Don't really understand how to manipulate the differential equation into (-1/ρ)(dρ/dT)

Especially the part where d[(m/ρ)]/(m/ρ) equals (-1/ρ)dρ

2. Jan 23, 2017

### Bystander

What are the properties of "m?"

3. Oct 23, 2017

### Chemesb

Given the equations you provided, we substitute V=m/ρ to find that β= (ρ/m)*(d/dT(m/ρ)). Then, take the derivative of 1/ρ with respect to temperature, so we don't just end up with -m/(ρ^2), we end up with -m/(ρ^2)*(dρ/dT). Mass is of course constant, and does not (noticeably) change with temperature. (Think back to implicit differentiation where we differentiate y^2 with respect to x to get 2y*y'). Now, we have β= (ρ/m)*(-m/[ρ][/2]*(dρ/dT)) which, when we cancel out a ρ and the m term, gives us β=(-1/ρ)(dρ/dT).

Last edited by a moderator: Oct 23, 2017