# Heat Capacity at Constant Volume or Pressure

Q:
I cannot understand something regarding the Cv and Cp of a liquid. Suppose to take the example of propane in liquid state.According to tables from the engineering site https://www.engineeringtoolbox.com/specific-heat-capacity-propane-Cp-Cv-isobaric-isochoric-d_2060.html we have that the specific heats are as follows:TempGas Gas Liquid Liquid Isobaric Isochoric Isobaric Isochoricï¿½C KJ/Kg/K KJ/Kg/K KJ/Kg/K KJ/Kg/K -251,49 1,28 2,37 1,5251,69 1,48 2,77 1,66401,75 1,55 3 1,77501,79 1,59 3,1 1,8Besides being a little bit bewildered from the fact that the liquid has two different heat capacities depending on keeping pressure or volume constant, and this bewilders me because the volume variation of the liquid is very tiny and thus not much energy is expelled as work comparing the two cases.I would like a confirmation on that because I am uncertain that there should be such a huge difference in specific heat for the liquid phase.On the other side I notice the strange coincidence that the heat capacity of the isobaric gas is the same as the heat capacity of the isochoric liquid and that may mean something and there should be a reason for that.What is the reason ?
- George Kourtis (age 54)
Athens Greece
A:

Great question!
In beginning courses we show the difference between Cp and Cv as simply coming from the work an ideal gas does as it expands. That's fine for ideal gases but it misses most of the story for liquids. (This very issue made for some interesting discussions as a group of us worked on the materials for a thermal physics course, because we had to learn it ourselves.)

Let's say you heat the liquid at fixed volume, measuring Cv. Not let it expand (or contract!) isothermally doing work (maybe negative!) until its internal pressure matches what the initial pressure was.You know for sure that the work done can't be the only term in the energy flow, because if it were then for liquids that happen to contract when heated (like water between 0° and 4° C) you'd end up with Cp < Cv. That would violate a theorem. For the liquid the internal energy of all the strongly interacting particles also depends strongly on volume. It's that change that gives most of the difference between Cp and Cv.

As for the match between Cp of the gas and Cv of the liquid, I think that's a coincidence, although I'm not sure.

Mike W.

(published on 12/15/2019)

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