(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that it follows from

[tex]\frac{dV}{V} = \beta dT[/tex]

that, for small changes, density has a similar linear relationship to thermal expansion, namely

[tex]\frac{d\rho }{\rho } = - \beta dT[/tex]

2. Relevant equations

3. The attempt at a solution

Well, at first glance this one seems ridiculously simple, but I tried setting up [tex]dV[/tex] as [tex]V_{1} - V[/tex], then inserting [tex]V_{1} = \frac{m}{\rho _{1}}[/tex] and similar for [tex]V[/tex], but when I go through the whole mathematical process that leaves me with the following equation:

[tex]\frac{d\rho }{\rho _{1}} = - \beta dT[/tex]

as opposed to:

[tex]\frac{d\rho }{\rho } = - \beta dT[/tex]

I'm not that good with LaTeX, so I haven't sketched the whole mathematical process I went through - there's not much to it anyway, to be honest - but does someone know what one needs to do to get the proper equation and where I went wrong? Is there some trick to it that I'm missing?

The final result for [tex]dV[/tex] is namely

[tex]\frac{-md\rho }{\rho \rho _{1}}[/tex]

and I was wondering if you could express [tex]\rho _{1} = \rho + d\rho [/tex], multiply and then dismiss the product [tex]\rho d\rho[/tex] as so small as to be irrelevant. Then you would namely get the right result, but I'm not that sure that product really is that small in comparison to the other one.

And sorry for any screw-ups in LaTeX, like I mentioned I'm not well versed in it (yet) and I needed almost half an hour to type even this up.

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# Homework Help: Derive the density relation to thermal expansion

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