Warm bath from hot and cold water: the mixing calculation
SNC2D Grade 10 Science · Climate Change
Before the class arrives, a science technician is preparing a 38.0 °C water bath for a class yeast-respiration experiment — warm enough to keep the yeast working, cool enough not to kill it, so the target is exact. On the prep-room counter: the hot-water reservoir, holding 2.40 kg at 70.0 °C; the cold tap, which runs at 15.0 °C once it has been left to steady; and the insulated bath tub itself, pre-warmed with a rinse so its own uptake can be ignored. The plan is a straight mix — pour the reservoir water, add cold tap water, and land on 38.0 °C exactly, no heater and no waiting. The question is how much tap water to add. And out of curiosity, the technician also wants to know how long the department's 300 W aquarium heater would have needed to do the same warming job on that tap water alone.
Every number in this problem is editable, the material included — change any value below and the whole chain recalculates.
- m_h = 2.4 kg — Hot water from the reservoir
- T_h = 70 °C — Reservoir temperature
- T_c = 15 °C — Cold tap temperature
- T_mix = 38 °C — Target bath temperature
- P = 300 W — Aquarium heater, for the comparison
- (a)the heat the reservoir water gives up cooling to the 38.0 °C target
- (b)the mass of 15.0 °C tap water that heat can raise to the same target
- (c)the time the 300 W heater would need for the same warming job
The hot side of the balance: the reservoir water's slide is 70.0 − 38.0 = 32.0 K. Keeping each stream's ΔT pinned to its own starting temperature is the whole bookkeeping of mixing — the hot water falls 32, the cold climbs 23, and swapping those two is the error that wrecks this calculation most often.
Carried onward at full precision, not this rounded figure.
In an insulated tub, heat lost equals heat gained, so the same Q now supplies the cold water's 23.0 K climb: m = Q/(cΔT) ≈ 3.34 kg. Water's c cancels top and bottom, which is why the answer is really just 2.40 × 32/23 — the masses balance in inverse proportion to their temperature swings, a seesaw with the pivot at 38.0 °C.
Carried onward at full precision, not this rounded figure.
The cold water's heat demand IS the hot water's 321.5 kJ — that equality was step 2's whole premise — so the heater comparison is t = E/P: nearly 18 minutes of a 300 W element to do what the pour does in seconds. Mixing moves no heat faster; it just relocates heat the reservoir already holds.
Carried onward at full precision, not this rounded figure.
Therefore the reservoir water surrenders 321.5 kJ on its way down to 38.0 °C, which lifts 3.34 kg of tap water up to the same mark — a 5.74 kg bath — and the 300 W heater would have ground away for 17.9 min to match what one pour accomplished.
Why this order
Mixing problems are a strict energy balance, and the chain keeps its two sides on separate lines on purpose: part (a) measures what the hot stream gives up, part (b) hands exactly that to the cold stream, and the join between them — heat lost equals heat gained — is stated once, where it can be examined. It is an assumption about the TUB (insulated, pre-warmed), not a law of algebra, and labs that skip the pre-warming discover their first batch lands a degree low because the tub itself drank part of the heat. The recurring student error is crossing the ΔTs: the hot water's 32 belongs to the hot mass, the cold water's 23 to the cold, and because water's specific heat cancels entirely, the surviving structure is a lever balance — m_hot × 32 = m_cold × 23, masses inversely proportional to their swings, with the target temperature as the pivot.
That lever is worth internalizing because it is how every mixing valve in a building thinks: a shower mixer, a boiler's tempering valve, a dairy's pasteurizer-cooler all solve part (b) continuously, in brass. Part (c) reframes the same joules as time and makes the energy tangible: 321.5 kJ is eighteen minutes of a 300 W element, which is why hot-water storage exists at all — a tank is a battery of already-delivered joules that a pour can release in seconds, at a rate no reasonable element could match live. Change the target to 40 °C and rerun the worksheet: the cold share drops to 2.88 kg, and the seesaw arithmetic shows why the last few degrees of a hotter bath cost disproportionately much cold-side capacity.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.