Mixing Temperature Calculator (final temperature)

Understanding the mixing temperature calculator — concept and principle

When two substances at different temperatures are mixed or placed in contact, heat flows from the hotter to the colder until they reach a common temperature. This calculator predicts that final temperature from each part's mass, specific heat and starting temperature. Typical uses are mixing bath or recipe water, quenching metal and calorimetry problems.

Like a seesaw, the heavier cold water pulls the balance point toward it: 44 °C.

How it is calculated

T_f = (m₁c₁T₁ + m₂c₂T₂) ÷ (m₁c₁ + m₂c₂)

m is mass in kg, c is specific heat in J/(kg·K), T is temperature in °C.
The products m·c are heat capacities (J/K): how much energy each part needs per degree.

The final temperature is therefore a weighted average of the starting temperatures, where each part's weight is its heat capacity.

The principle

Energy is conserved. If no heat escapes and nothing melts or boils, the heat lost by the hot part equals the heat gained by the cold part: m₁c₁(T₁ − T_f) = m₂c₂(T_f − T₂). Solving this single equation for T_f gives the formula above. Because only temperature differences appear, the result is the same whether you work in °C or K. The final temperature always lies between the two starting temperatures and sits closer to the part with the larger heat capacity.

Worked example

Inputs: part 1 is 2 kg of water (c = 4186 J/kg·K) at 80 °C; part 2 is 3 kg of water (c = 4186) at 20 °C.
Heat capacities: 2 × 4186 = 8372 J/K and 3 × 4186 = 12,558 J/K; total 20,930 J/K.
Numerator: 8372 × 80 + 12,558 × 20 = 669,760 + 251,160 = 920,920.
T_f = 920,920 ÷ 20,930 = 44 °C.
Since both parts are water, c cancels and T_f = (2 × 80 + 3 × 20) ÷ 5 = 44 °C.
Check: the hot water loses 8372 × 36 = 301,392 J and the cold water gains 12,558 × 24 = 301,392 J.

Each point is the calculator re-run with only “Temp 1 (°C)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Temp 1 (°C)	Final temperature (°C)
20	20
40	28
60	36
80	44
100	52
120	60
160	76

How to read the result

The result is an upper bound on how close to "perfectly mixed" you get without losses. In practice the container absorbs some heat and the mixture loses heat to the air, so a measured temperature is usually a little lower than predicted when mixing into a cool container. A high-specific-heat substance such as water dominates the outcome: a hot steel part dropped into water barely warms the water, because steel's c (about 490 J/kg·K) is less than one eighth of water's.

Common mistakes

Forgetting the container; for accurate calorimetry add its heat capacity as a third term.
Using the formula when ice melts or water boils; the latent heat must be included separately.
Mixing units of c between the two parts (J/kg·K for one, kJ/kg·K for the other).

Keep in mind: The formula assumes no heat loss, no phase change, no chemical reaction and constant specific heats. Treat the result as an ideal estimate.

Frequently asked questions

Can I mix more than two substances?

Yes. Extend the sums: T_f = ΣmᵢcᵢTᵢ ÷ Σmᵢcᵢ, adding one term per part.

How much cold water do I need to reach a target temperature?

Rearrange: m₂ = m₁c₁(T₁ − T_f) ÷ (c₂(T_f − T₂)). For 2 kg at 80 °C cooled to 40 °C with 20 °C water, m₂ = 2 × 40 ÷ 20 = 4 kg.

Does it matter if I use litres instead of kilograms for water?

For water near room temperature 1 L ≈ 1 kg, so the error is small; for hot water the density is a few percent lower.

Sources: D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, Wiley, chapter on temperature, heat and the first law

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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