At a birthday party, a student ate a piece of cake (food energy value of 400 calories). To prevent this energy from being stored as fat, she decides to drink ice water at 0 C. She reasons that the ingested ice water will be warmed up to her normal body temperature of 37 C and absorb the energy. How much ice water would she have to drink to absorb the energy generated by metabolizing the birthday cake?

400 Cal = 1674.72 J

Q =c•m•ΔT,
m = Q/c•ΔT = 1674.72/4180•37 =0.01 kg

To answer this question, we need to understand the concept of specific heat capacity and the energy required to raise the temperature of a substance.

1. First, let's calculate the energy required to raise the temperature of the ice water from 0°C to 37°C.
The specific heat capacity of water is 4.18 J/g°C (joules per gram per degree Celsius).
Let's assume the ice water initially weighs "x" grams.

The energy required can be calculated using the formula:
Energy = mass * specific heat capacity * change in temperature

Change in temperature = (37°C - 0°C) = 37°C

Energy = x grams * 4.18 J/g°C * 37°C

2. The next step is to convert this energy from joules to calories.
1 calorie (cal) = 4.18 joules (J)

Energy in calories = Energy in joules / 4.18

3. Now that we have the energy required to raise the temperature of the ice water from 0°C to 37°C, we can calculate the mass of ice water needed to absorb the energy from the birthday cake.
Given that the food energy value of the cake is 400 calories, we equate it to the energy in calories calculated in step 2.

Energy from the cake = Energy in calories

Ice water mass (x grams) * 4.18 J/g°C * 37°C = 400 calories * 4.18 J/cal

4. Finally, we can solve for the mass of ice water, x.

x grams = (400 calories * 4.18 J/cal) / (4.18 J/g°C * 37°C)

Simplifying this equation will give us the mass of ice water required to absorb the energy generated by metabolizing the birthday cake.

Please note that this calculation assumes that all the energy from the cake will be absorbed by the ice water, neglecting any energy loss due to metabolic processes.