Density Change Caused by Temp. Change

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[SOLVED] Density Change Caused by Temp. Change

Homework Statement


Show that the change in the density [itex]\rho[/itex] of a substance, when the temperature changes by [itex]\Delta T[/itex], is given by [itex]\Delta \rho = - \beta \rho \Delta T[/itex].


Homework Equations


[tex]\rho = \frac{m}{V}[/tex]
[tex]\Delta V = \beta V \Delta T[/tex]


The Attempt at a Solution


Let [itex]\rho_1[/itex] and [itex]\rho_2[/itex] be the densities of the substance before and after the temp. change [itex]\Delta T[/itex]. After some algebra, I get:

[tex]\Delta \rho = \rho_2 - \rho_1 = - \beta \rho_2 \Delta T[/tex]

This differs from that given in the problem statements because [itex]\rho[/itex] is [itex]\rho_1[/itex] in the above. I guess this must be typo right?
 
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Answers and Replies

  • #2
Kurdt
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What are the explicit densities before and after the increase in temperature? That is the key to getting this and a lot of people make a mistake. The density before is simple:

[tex] \rho_0=\frac{m}{V_0} [/tex]

but the final density is:

[tex] \rho = \frac{m}{(V_0+\Delta V)} [/tex]
 
  • #3
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Are you implying that I made a mistake? I have checked the algebra twice and I still get the result I got. If I use what you wrote for the densities, I still get the same result.
 
  • #4
Kurdt
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Ahh ok I see what you've done, my mistake. Yes I get the same answer as you so I'd wager it was a spelling error unless we both made the same mistake.
 

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