Ice-water mass ratio is maintntained as `1:1` in a given system containing water in equilibrium with ice at constant pressue . If `C_(p)` (ice) =`C_(p)` (water) =4.18 J `mol^(-1)K^(-1)` molar heat capacity of such a system is :

Options

A. zero
B. Infinity
C. `4.182 JK^(-1)mol^(-1)`
D. `75.48JK^(-1)mol^(-1)`

Answer

To solve the problem, we need to determine the molar heat capacity of a system containing ice and water in equilibrium at a constant pressure, where the mass ratio of ice to water is 1:1. Given that the molar heat capacities of both ice and water are equal to 4.18 J mol⁻¹ K⁻¹, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a system where ice and water are in equilibrium. This means that the rate of melting of ice is equal to the rate of freezing of water. Therefore, there is no net change in the amount of ice or water over time.

2. **Equilibrium Condition**:
- Since the system is at equilibrium, the temperature remains constant. This implies that any heat absorbed by the ice to melt into water is equal to the heat released by the water as it freezes into ice.

3. **Heat Capacity Definition**:
- The molar heat capacity at constant pressure (Cₚ) is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Kelvin (or Celsius).

4. **Change in Temperature**:
- In this scenario, since the system is at equilibrium, there is no change in temperature (ΔT = 0).

5. **Calculating Molar Heat Capacity**:
- The formula for molar heat capacity at constant pressure is given by:
\[ C_{p} = \frac{Q}{\Delta T} \]
- Where Q is the heat exchanged. Since ΔT = 0, substituting this into the equation gives:
\[ C_{p} = \frac{Q}{0} \]
- This leads to an undefined situation, which in thermodynamic terms indicates that the molar heat capacity approaches infinity.

6. **Conclusion**:
- Therefore, the molar heat capacity of the system, where ice and water are in equilibrium at constant pressure, is effectively infinite.

### Final Answer:
The molar heat capacity of the system is infinite.