How many calories of heat are gained to turn 10 grams of ice at 0 C into water at 0 C?

To determine the number of calories of heat gained to turn ice at 0°C into water at 0°C, we need to consider two steps: raising the temperature of ice to its melting point and then melting the ice.

Step 1: Raising the temperature of ice to 0°C.
The specific heat capacity of ice is 0.5 calories/gram °C. Therefore, we can calculate the amount of heat required to raise the temperature of 10 grams of ice from -273°C (absolute zero) to 0°C using the formula:

Heat = mass × specific heat capacity × temperature change

Heat = 10 grams × 0.5 calories/gram °C × (0°C - (-273°C))
= 10 grams × 0.5 calories/gram °C × 273°C
= 1365 calories

Step 2: Melting the ice.
The heat of fusion for ice is 79.7 calories/gram. So, we need to calculate the amount of heat required to convert 10 grams of ice into water at 0°C:

Heat = mass × heat of fusion

Heat = 10 grams × 79.7 calories/gram
= 797 calories

Now, to find the total heat gained, we add the results from step 1 and step 2:

Total heat gained = 1365 calories + 797 calories
= 2162 calories

Therefore, it takes 2162 calories of heat to turn 10 grams of ice at 0°C into water at 0°C.