Density Change Caused by Temp. Change

In summary, the change in density of a substance when the temperature changes by \Delta T is given by \Delta \rho = - \beta \rho \Delta T. This is derived by considering the densities before and after the temperature change, \rho_1 and \rho_2 respectively. It is important to note that the final density, \rho_2, is calculated as \rho_2 = \frac{m}{V_0 + \Delta V}. Any discrepancy in the given answer is likely due to a spelling mistake.
  • #1
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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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  • #2
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
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
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.
 

1. How does temperature affect the density of a substance?

As temperature increases, the density of a substance decreases. This is because as temperature increases, the molecules of the substance have more energy and move faster, causing them to spread out and take up more space, leading to a decrease in density.

2. Is there a specific formula to calculate density change caused by temperature change?

Yes, the formula is: Density at a certain temperature = Density at a reference temperature x (1 + coefficient of thermal expansion x (temperature - reference temperature)). The coefficient of thermal expansion is a constant value specific to each substance.

3. Can density change caused by temperature change be reversed?

Yes, as temperature decreases, the density of a substance increases. This is because as temperature decreases, the molecules of the substance have less energy and move slower, causing them to pack closer together and take up less space, leading to an increase in density.

4. How does the density change of a substance due to temperature affect its buoyancy?

If the density of a substance decreases due to an increase in temperature, it will become less dense than the surrounding liquid or gas and will float. However, if the density increases due to a decrease in temperature, it will become more dense than the surrounding liquid or gas and will sink.

5. Is density change caused by temperature change the same for all substances?

No, the density change caused by temperature change varies for different substances. The coefficient of thermal expansion, which determines how much a substance's density changes with temperature, is different for each substance. Some substances may become less dense with an increase in temperature, while others may become more dense.

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