0
$\begingroup$

I want to calculate the density distribution inside a column of a solid material of which I know all properties (density, compressibility) at the top (room pressure). Using the definition of isothermal compressibility as: $$ \beta_T = \frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T $$ and the definition of density as $\rho = \frac{m}{V}$, I arrive at $$ \left( \frac{\partial \rho}{\partial P} \right)_T = \rho \beta_T $$ Integrating this differential equation I get an exponential function for density $\rho = \rho_0 \exp(\beta P)$. However, doing so I assumed that $\beta_T$ does not depend on pressure. But if I write: $$ \frac{\partial \beta_T}{\partial P} = \frac{\partial}{\partial P} \left( \frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T \right) $$ it looks to me like you could find an expression for this involving only stuff we already know. It might be a bit nit-picky, since the pressure depence might be small, but if the pressure becomes large enough, it could make a difference?

$\endgroup$
2
  • 1
    $\begingroup$ You ask about $\partial\beta/\partial P$ but your title suggests a derivative w.r.t. temperature. $\endgroup$
    – Kyle Kanos
    Commented Jan 11, 2016 at 17:44
  • $\begingroup$ I changed the title to "... depend on pressure". Thanks for pointing this out! $\endgroup$ Commented Jan 12, 2016 at 8:29

1 Answer 1

0
$\begingroup$

If you want the density distribution as a function of height in 'typical' column of solid building material here on Earth, then to a very good approximation it will simply be a linear function of pressure as given by your isothermal compressibility equation AND it will be very, very small (hence, a linear approximation is all you need. Forget about the exponential function of pressure you wrote - you will never get there.).

As a reality check, consider the water at the bottom of the Mariana Trench which at 36,000 feet or 6.8 miles down is the deepest spot in the oceans. The bulk modulus of water is about 23 kbars (23,000 atm), and the pressure there is about 1 mbar (1000 atm), so the density of water there is around 5% higher than water at the atmospheric pressure at the same temperature. Of the tallest buildings on Earth, Burj Khalifa is the tallest with a height of about 2,700 feet, or less than 1/10 the depth of the Mariana trench. So the linear approximation for the relationship between pressure and density (or pressure and volume) is all you need for any current man-made structure.

If you want to consider what happens to the pressure-density relationship at higher pressures when a linear relationship no longer holds, then the first thing you need to do is to discard the assumption that the isothermal compressibility $\beta_T$ has no pressure dependence. Instead, you have to use something like the non-linear Birch-Murnaghan equation of state, which is used by scientists studying the equations of state of materials under extreme pressure conditions using either diamond anvil cells or shock waves.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.