Molar heat capacity of water at equilibrium

Question

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?

Answer 1

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.