# State of water and max pressure

1. Sep 1, 2008

### zoki85

Imagine a hollow sphere made of perfect material.
Its walls are perfectly resistant to any thermal and mechanical stress and they can't be distorted.Suppose the interior of the sphere is completely filled with water.The whole system is at room temperature and standard atmospheric pressure.
If you cool down the water enough ,what will be the highest presure the sphere will be exposed to?Any ideas?I don't know even how to estimate this.

2. Sep 1, 2008

### tiny-tim

Hi zoki85!

Are you allowed to cool it below 0ºC?

3. Sep 1, 2008

### Count Iblis

This is an interesting problem. If you cool the water, the water contracts. The water then evaporates, so that the whole volume is filled with water and water vapor. The pressure inside the sphere is thus simply the vapor pressure of water at the temperature of the system.

If you cool to low enough temperatures, the pressure will increase again. Below 0°C at 1 bar you would get ice, but the volume would be larger than the volume of the sphere. So, what you'll get instead is water under high pressure.

4. Sep 1, 2008

### granpa

what are you studying?

5. Sep 1, 2008

6. Sep 2, 2008

### zoki85

To ~0 K all the way down if necessary !
I'm looking for max. presure attainable within temperature range<300 K.

7. Sep 2, 2008

### Count Iblis

You can try to roughly estimate this using the properties of ice and water. The isothermal compressibility of ice at 0°C is about 0.13 GPa^-1, the (volumetric) thermal expansion coeficient is about 166·10^-6 K^-1. Melting point drops approximately by 74 K per GPa of pressure. Now, the compressibility of water is larger than the compressibility of ice, it is 0.51 GPa^(-1) at 0°C.

So, this suggests (but you have to look at this in more detail), that you should cool to exactly that temperature at which you just get ice in the sphere. The volume is reduced by approximately 10%, which corresponds to a pressure of about 7.7 10^8 Pa if the ice were to stay ice. That's of course not true at 0 °C, but if we reduce the temperature to 273 - 74 K * 0.77 = -57 °C it should stay ice. At that temperature the ice has a slightly larger density at 1 bar pressure than at 0 ° C, but that's insignificant, it doesn't change the estimate of 7.7 10^8 Pa pressure needed to compress the ice by a factor 1.1 (obtained using the compressibility at 0 °C, but I don't think using conpressibility at -57 is that much different).

If we lower the temperature further, the density will increase, so the ice would be less compressed and thus the pressure would be less. At higher temperatures, there will be water in the sphere which has a significantly higer compressibility, so the pressure would be much lower.

So, I think the maximum pressure would be roughly
7.7 10^8 Pa = 7,700 bar at -57 °C

8. Sep 4, 2008

### zoki85

This is a good way of looking at the problem Count Iblis.