Question

Ice-water mass ratio is maintained as $1:1$ in a given system containing water in equilibrium with ice at constant pressure. If ${{C}_{p}}\left( ice \right)={{C}_{p}}\left( water \right)=4.18Jmo{{l}^{-1}}{{K}^{-1}}$ molar heat capacity of such a system is
$\begin{align}
  & \text{A}\text{. Zero} \\
 & \text{B}\text{. Infinity} \\
 & \text{C}\text{. }4.182Jmo{{l}^{-1}}{{K}^{-1}} \\
 & \text{D}\text{. 75}\text{.48}Jmo{{l}^{-1}}{{K}^{-1}} \\
\end{align}$

Answer 1

Hint: Heat capacity of a system is the amount of heat required to raise the temperature of the system by one degree celsius. Express the heat capacity mathematically in terms of the heat given and the change in temperature of the system. Put the values for the given system in the question to find the required answer.

Complete step by step answer:
Molar heat capacity of a substance can be defined as the amount of heat or the amount of energy required to raise the temperature of one mole of the substance by a temperature of one degree Celsius.
We can define,
$\begin{align}
  & q=C\Delta T \\
 & C=\dfrac{q}{\Delta T} \\
\end{align}$
Where, C is the heat capacity, q is the heat supplied and $\Delta T$ is the change in temperature of the system.
Given in the question that, the mass ratio of ice and water is $1:1$ . The molar heat capacity of ice and water at constant temperature are given as,
${{C}_{p}}\left( ice \right)={{C}_{p}}\left( water \right)=4.18Jmo{{l}^{-1}}{{K}^{-1}}$
If we transfer some heat or energy to ice, the ice will melt to water. But because of the latent heat of fusion, there will not be a change in temperature of the system.
Now, in the system given in the question, both ice and water are there in the same amount and are in equilibrium with each other. So, if we transfer some energy to the ice, the ice will melt to water but the temperature will remain the same because of the latent heat of fusion. Again, the system is in equilibrium. So, the amount of ice transforming to water will be equal to the amount of water transforming to ice. So, the temperature of the system will remain constant and the change in temperature of the system will be zero.
So, the molar heat capacity of the system will be,
$C=\dfrac{q}{\Delta T}=\dfrac{q}{0}=\infty $
So, the molar heat capacity of the system will be infinite.
The correct option is (B).

Note:
The change in temperature of the system is zero due to the latent heat. When the ice transforms to waters its phase changes and the applied heat is used in this phase change. So, no change in temperature will happen. This heat is called the latent heat.