# Density Change Caused by Temp. Change

[SOLVED] Density Change Caused by Temp. Change

## Homework Statement

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

## Homework Equations

$$\rho = \frac{m}{V}$$
$$\Delta V = \beta V \Delta T$$

## 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:

Kurdt
Staff Emeritus
Gold Member
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:

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

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.

Kurdt
Staff Emeritus