Change in Density due to Temperature and Pressure Change

[JSGandora]

Homework Statement

Estimate the percent change in density of iron when it is still a solid, but deep in the Earth where the temperature is 2000^\circ C and it is under 5000 atm of pressure. Take into account both thermal expansion and changes due to increased outside pressure. Assume both the bulk modulus and the volume coefficient of expansion do not vary with temperature and are the same as at normal room temperature. The bulk modulus for iron is about 90\times10^9N/m^2.

We also know that the volumetric expansion coefficient of iron is 35\times10^{-6}(^{\circ}C)^{-1}.

Also, 1atm=1.103\times10^5N/m^2.

Homework Equations

\frac{\Delta V}{V_0}=-\frac{1}{B}\Delta P where B is the bulk modulus.
\Delta V=V_0\beta\Delta T where \beta is the volumetric expansion coefficient.

The Attempt at a Solution

I'm assuming the change in volumes adds up so we have using the two equations successively, we have
\frac{\Delta V}{V_0}=\left(-\frac{1}{90\times10^9}\right)\left(5000atm\right) \left(\frac{1.103\times10^5N/m^2}{1atm}\right)=-0.00563
and
\frac{\Delta V}{V_0}=(4975^\circ C)(35\times10^-6\left(^\circ C\right)^{-1}=0.17413

So the total change in volume is 0.17413-0.00563=0.168495.

Then we could easily get the percent change in density. However, I am unsure if I am applying this correctly. Am I allowed to add the change in volumes?

 

[ehild]

However, I am unsure if I am applying this correctly. Am I allowed to add the change in volumes?

The volume becomes (1+ΔV/V₀) times the initial one in both processes, and you need to multiply them when there is change of volume both because of pressure and temperature. When ΔV/V₀<<1, you can simply add the relative changes (with the proper signs) but 0.17 is not very small compared to unity. ehild

 

[JSGandora]

Wait, so it should be (0.17413)(-0.00563)=-0.0009803519? That doesn't make intuitive sense to me. Why do we multiply the change in volumes?

 

[ehild]

Wait, so it should be (0.17413)(-0.00563)=-0.0009803519?

NO. We multiply the volume ratios. V/V₀=1.17413*(1-0.00563)=1.16752. ΔV/V₀=0.16752.

ehild

 

[JSGandora]

Oh, I see now. Could you explain why it must be multiplied? I don't understand.

 

[ehild]

The formulae for volume change due to pressure and temperature refer to cases when the other quantity is constant. Pretend you put the piece of iron at high pressure first, when the volume decreases by Vo(ΔP/B). The new volume, V₁=Vo(1-ΔP/B) is warmed up, and the new volume serves as "Vo" in thermal expansion. But the results are almost the same, and the the formulae are approximations anyway. So both ways should be accepted for the relative change of the volume, which is 0.168 with 3 significant digits by both methods.

ehild

 

[JSGandora]

Oh, thanks you so much. I understand now.