I am working through the questions in Zemansky's Heat and Thermodynamics.

Question 2.8 says:

The thermal expansivity and the compressibility of water are given in the accompanying table. Draw a graph showing how $\left(\partial{P} / \partial{\theta}\right) _V$ depends on the temperature. If water were kept at constant volume and the temperature were continually raised, would the pressure increase indefinitely?

t, Celcius                 0    50   100   150   200   250   300 
Expansivity (10^-3)        0   .45   .74  1.02  1.35  1.80  2.90
Compressibility (10^-4)  .45   .45   .50   .62   .85  1.50  3.05 

I can work out the required partial (it is expansivity divided by compressibility) and it's clearly positive everywhere in the range 0-300C. So pressure will increase with temperature over that range, at constant volume. But indefinitely?

A common sense answer is that at a certain point it will become very hard to raise the temperature, or that any containing vessel will burst / melt at a sufficiently high temperature, but the question seems to ask what would happen if the temperature could keep being raised without the volume changing.

There is nothing I can find earlier in the text which gives a direct answer -- there is a phase diagram for water but it doesn't go much beyond 300C.

  • $\begingroup$ $\left(\partial{P} / \partial{\theta}\right) _V$ might be positive but is it decreasing or increasing as the temperature increases above $300^\circ\rm C$? $\endgroup$ – Farcher Jun 25 '17 at 9:20
  • $\begingroup$ The derivative is decreasing as temperature approaches 300C, but still positive. It looks like there is an inflection point around 250C so I'm not sure if the derivative would ever become zero or negative (which is what would be needed for pressure not to rise indefinitely). I have seen a few charts in other documents which suggest that isochores flatten out at higher temperatures, but are still slightly upward sloping at the end of the charted range. Unfortunately I can't find charts which go beyond 1000K or so. $\endgroup$ – Tony Jun 25 '17 at 9:26
  • $\begingroup$ From the data in the table, you wouldn't be able to conclude whether or not the pressure will increase indefinitely. From statistical mechanics point of view, pressure is because collision of molecules to the wall and thus will be a function of the molecules' kinetic energy, which is its temperature. Thus the pressure will increase indefinitely based on that description. $\endgroup$ – user115350 Jun 25 '17 at 17:00

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