Question

How much boiling water would you need to raise the bath to body temperature (about 37 ∘C)? Assume that no heat is transferred to the surrounding environment.

Answer 1

Let's find out how much boiling water is necessary to raise the temperature of each litre from 25 C water to 37 C.

Heat lost by hot water = heat gained by cold water:

$$\Delta \mathrm {H} _ {\mathrm {c}} \text { hot water} = - \Delta \mathrm {H} _ {\mathrm {h}} \text { cold water}$$ $$\mathrm {m} _ {\mathrm {c}} \mathrm {C} \Delta \mathrm {T} _ {\mathrm {c}} \text { cold water} = - \mathrm {m} _ {\mathrm {b}} \mathrm {C} \Delta \mathrm {T} _ {\mathrm {b}} \text { hot water}$$

The difference in heat capacities is negligible and can be canceled out:

$$\mathrm {m} _ {\mathrm {c}} \Delta \mathrm {T} _ {\mathrm {c}} = - \mathrm {m} _ {\mathrm {b}} \Delta \mathrm {T} _ {\mathrm {b}}$$ $$\mathrm {m} _ {\mathrm {b}} / \mathrm {m} _ {\mathrm {c}} = \Delta \mathrm {T} _ {\mathrm {c}} / \Delta \mathrm {T} _ {\mathrm {b}}$$ $$\mathrm {m} _ {\mathrm {b}} / \mathrm {m} _ {\mathrm {c}} = 12 / 63 = 0.19 \approx 0.2$$

So, for each liter of cold water we need 0.2 l of boiling water.

If the amount of water in bath is 150 l then you need

$$\mathrm {m} _ {\mathrm {b}} = 0.2 \mathrm {m} _ {\mathrm {c}} = 30 \mathrm {l} \text { of boiling water.}$$