How does density change with depth? How does the density of water with constant temperature change with depth?
Is this even possible to calculate? I know that density at the bottom is higher than at the top, but I am interested in theoretical numbers - by how much?
 A: The bulk modulus of water is $2\times 10^9$ Pa $= 2\times 10^4$ ba $= 1.97\times 10^4$ atm.
The bulk moudulus is defined as the additional Pressure devided by the corresponding rate in volume change:
$$
   B = -V\frac{\Delta P}{\Delta V} =  -\frac{\Delta P}{\Delta V /V}.
$$
Therefore for a increse in the pressure $\Delta P$ leads to a change in density:
$$
  \rho = \frac{M}{V}.\\
  \Delta \rho = -  \frac{M}{V^2} \Delta V = - \rho \frac{\Delta V}{V}.\\
\frac{\Delta \rho}{\rho} = - \frac{\Delta V}{V} = \frac{\Delta P}{B}
$$
Consider a pond of water 100 meters in depth, The bottom of pond has an addition pressure than that of the surface.
$$
\Delta P = \rho g h = 1 \times 980 \times 10000 \approx 10^7 dyne/cm^2 \approx 9.8 ba. 
$$
The cooresponding change in density
$$
\frac{\Delta \rho}{\rho} = \frac{\Delta P}{B} = \frac{9.8}{2\times 10^4}\approx 0.05 \%
$$
Edited by OzzieO:
I think all the physics is right. It might not make any difference but my guess is this is probably an adiabatic relationship and not isothermal, which was in the original question. In any case I suspect the math is off by a decimal place and the correct answer is 0.05%. Water is pretty incompressible. The density change for 500m of seawater is about 0.2% using the same physics.
A: Water is relatively incompressible but nonetheless not absolutely. Its compressibility depends on its phase as well as temperature and pressure. When you say change with depth, I assume you mean increasing pressure.
The way I would think of your description is some volume of liquid water wiith a heat bath beneath it to keep it at constant temperature and then a piston applied at the top to start simulating "depth" i.e. increasing pressure. Below is a link to a PT diagram for water. You can see that starting from some liquid phase at constant temperature you eventually always reach a solid (ice) phase at high enough pressure.
PT diagram water
Practical calculations of all this can probably be found in engineering references which I am not familiar.
