# Density Change Caused by Temp. Change

1. May 17, 2007

### e(ho0n3

[SOLVED] Density Change Caused by Temp. Change

1. The problem statement, all variables and given/known data
Show that the change in the density $\rho$ of a substance, when the temperature changes by $\Delta T$, is given by $\Delta \rho = - \beta \rho \Delta T$.

2. Relevant equations
$$\rho = \frac{m}{V}$$
$$\Delta V = \beta V \Delta T$$

3. The attempt at a solution
Let $\rho_1$ and $\rho_2$ be the densities of the substance before and after the temp. change $\Delta T$. After some algebra, I get:

$$\Delta \rho = \rho_2 - \rho_1 = - \beta \rho_2 \Delta T$$

This differs from that given in the problem statements because $\rho$ is $\rho_1$ in the above. I guess this must be typo right?

Last edited: May 18, 2007
2. May 18, 2007

### Kurdt

Staff Emeritus
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:

$$\rho_0=\frac{m}{V_0}$$

but the final density is:

$$\rho = \frac{m}{(V_0+\Delta V)}$$

3. May 18, 2007

### e(ho0n3

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. May 18, 2007

### Kurdt

Staff Emeritus
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.