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Express volume expansivity in terms of density

  1. Jan 22, 2017 #1
    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. jcsd
  3. Jan 23, 2017 #2

    Bystander

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    What are the properties of "m?"
     
  4. Oct 23, 2017 #3
    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: Oct 23, 2017
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