# Molar heat capacity of water at equilibrium

Molar heat capacity is defined as the heat energy given to one mole of substance to raise its temperature by 1 degree celcius (or 1 Kelvin)

Mathematically, $$C_{m} = \frac {q}{n\Delta T}$$

Here's a question on Molar heat capacity:

Molar heat capacity of water in equilibrium with ice at constant pressure is : a) 0 b) infinity c) 40.45 kJ/K/mol d) 78.5 J/K/mol

The answer is given as infinite. However, shouldn't it be impossible to determine, since if we provide heat to the system, it's temperature also changes? Can someone please explain to me why it is infinite?

• Yeah, you're right. Sep 24, 2023 at 1:15
• Because you have a phase change going on: Sep 24, 2023 at 2:13
• IMHO, it is badly formulated question. Heat capacity of the system with water as substance is infinity due the phase change. But molar heat capacity of water as liquid phase remains the same as the one at 0 Deg C, roughly 4.2x18 J/K/mol. They explicitly ask about water as the liquid phase, in contact with ice. Sep 24, 2023 at 5:53

So water is at equilibrium with ice at 273.15K or 0 degrees Celsius.

The difference between ice and water both at 0 degrees Celsius is evidently not their temperature (they’re both at 0 degrees). Yet there is clearly a difference in energy between the two species.

This difference arises as ice is lower in energy as it has formed bonds to make a solid packing structure without changing temperature. In other words when at the phase equilibrium temperature, the energy supplied/released is being used to form/break intermolecular bonds, not to change the temperature.

This can also be seen on a temperature vs phase diagram, when at the temperature of a phase equilibrium, this corresponds to the ‘flat line’ on this type of a graph.

Therefore the answer is zero, if you supplied energy to water at the phase equilibrium with ice then it would not change temperature, as all the energy would be used to break bonds, not change temperature.

Of course if you supplied an excess energy then the temperature would change, but in doing so the species would no longer be in the temperature relevant to the phase equilibrium any longer.

• Thank you for the clarification, @Kirandk. The book has evidently marked the wrong option as correct, so I thought that the correct option is zero. I'll edit the question now. Sep 24, 2023 at 2:41
• The molar heat capacity of water in equilibrium with ice is the same as molar heat capacity of water without ice presence. The flaw of the question formulation is confusing water(I) ( the liquid phase of substance water(II)) with water(II). Water(II) cannot be in equilibrium with ice as the solid phase of water(II). Water(I) can be. Sep 24, 2023 at 9:07