How much boiling water would you need in order to raise the bath to body temperature (about 37)? Assume that no heat is transferred to the surrounding environment. Imagine that your water heater has broken, but you want to take a bath. You fill your bathtub with 25kg of room-temperature water (about 25). You figure that you can boil water on the stove and pour it into the bath to raise the temperature.

[mass boiling water x specific heat x (Tfinal-Tinitial)] + [mass tub water x specific heat x (Tfinal-Tinitial)] = ??

I expect you will be surprised at the answer.

What is the mass of boiling water and the mass of tub water?

To determine how much boiling water you would need to raise the bath to body temperature (about 37°C), we can use the equation for specific heat capacity:

Q = m * c * ΔT

Where:
Q is the heat energy required
m is the mass of the water
c is the specific heat capacity of water
ΔT is the change in temperature

Let's break down the problem step by step:

Step 1: Determine the change in temperature
The desired change in temperature is from room temperature (25°C) to body temperature (37°C), so ΔT = 37 - 25 = 12°C.

Step 2: Calculate the heat energy required to raise the temperature of the existing water.
Using the specific heat capacity of water, which is approximately 4.18 J/g°C, and the mass of the water (25 kg), we can calculate the heat energy required:

Q1 = m * c * ΔT
= 25 kg * 4.18 J/g°C * 12°C

Step 3: Calculate the heat energy that boiling water would release when it cools down to the bath temperature (37°C).
When boiling water is poured into the bath, it will cool down to the bath temperature. The heat energy released by the boiling water should be equal to the heat energy absorbed by the existing water to raise its temperature. Therefore, we can calculate it as follows:

Q2 = Q1

Step 4: Calculate the mass of boiling water required.
We already know the specific heat capacity of water (4.18 J/g°C) and the change in temperature (12°C). To solve for the mass of boiling water (m), we rearrange the equation:

Q2 = m * c * ΔT

m = Q2 / (c * ΔT)

Substituting the calculated Q2 value from Step 3 and the values for c and ΔT:

m = Q2 / (4.18 J/g°C * 12°C)

Finally, by calculating m, we can determine the mass of boiling water required to raise the bath to body temperature (about 37°C).