Warm bath from hot and cold water: the mixing calculation
SNC2D Grade 10 Science · Climate Change
A science technician is preparing a 38.0 °C water bath for a class yeast-respiration experiment. The urn holds 2.40 kg of water at 70.0 °C; the cold tap runs at 15.0 °C; the insulated bath tub itself is pre-warmed, so its own uptake can be ignored. The technician wants to know how much tap water to add so the mix lands on 38.0 °C exactly — and, out of curiosity, how long the department's 300 W aquarium heater would have needed to do the same warming job on that tap water alone.
- m_h = 2.4 kg — Hot water from the urn
- T_h = 70 °C — Urn 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 urn 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 ledger: the urn 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 funds 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 warming bill 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 moves heat that was already paid for at the urn.
Carried onward at full precision, not this rounded figure.
Therefore the urn 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 double-entry bookkeeping, and the chain keeps the two entries on separate lines on purpose: part (a) prices what the hot stream gives up, part (b) spends exactly that on 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 ledger. 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-paid-for joules that a pour can withdraw 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.