Introduction:
When 1 kg of ice at 0°C melts to water at 0°C, there is a change in entropy. In order to calculate this change, we need to consider the latent heat of ice.

Explanation:
1. Definition of Entropy:
Entropy is a measure of the disorder or randomness in a system. It is a thermodynamic property that increases with the degree of disorder.

2. Change in Entropy during Melting:
During the process of melting, the ice undergoes a phase change from a solid to a liquid. This phase change is accompanied by an increase in entropy. The change in entropy can be calculated using the formula:
ΔS = Q/T
where ΔS is the change in entropy, Q is the heat transferred, and T is the temperature.

3. Calculation of Heat Transferred:
The heat transferred during the melting process can be calculated using the latent heat of ice. The latent heat of ice is the amount of heat required to change 1 kg of ice at 0°C to water at 0°C. In this case, the latent heat of ice is given as 80 cal/°C.

4. Calculation of Change in Entropy:
Using the formula ΔS = Q/T, we can substitute the values. Here, Q is the latent heat of ice, which is 80 cal/°C, and T is the temperature, which is 0°C. Therefore, the change in entropy is:
ΔS = 80 cal/°C / 0°C = 80 cal/°C

5. Conversion to Kelvin:
To convert the change in entropy to kelvin, we need to add the conversion factor of 273.15. Therefore, the change in entropy is:
ΔS = 80 cal/°C + 273.15 = 353.15 cal/K

Conclusion:
The resulting change in entropy when 1 kg of ice at 0°C melts to water at 0°C is 353.15 cal/K. Therefore, the correct option is (b) 293 cal/K.

b