Last updated: 2026-06-15

Final Temperature of a Mixture

Pour hot coffee into cold milk and the mixture settles somewhere in between. Exactly where depends not just on the two temperatures but on how much of each substance there is and how stubbornly each resists temperature change. This calculator finds that equilibrium temperature from a heat balance, and reports how much energy crosses from the hot side to the cold.

The principle

Inside an insulated container no heat escapes, so the heat lost by the hotter substance equals the heat gained by the cooler one:

Here $m$ is mass, $c$ is specific heat capacity, and $T_f$ is the shared final temperature. Solving for $T_f$ gives a weighted average:

Each substance is weighted by the product $mc$ — its heat capacity. The final temperature always lands between the two starting values, pulled toward whichever substance has the greater heat capacity.

Worked example

Mix $0.3\ \text{kg}$ of water at $90\ ^\circ\text{C}$ with $0.2\ \text{kg}$ of water at $20\ ^\circ\text{C}$. Both are water, so $c$ cancels and the result is a mass-weighted mean:

The heat that flows from the hot water into the cold is

Common specific heats

Water's unusually high value is why it dominates most everyday mixtures: it takes far more energy to shift its temperature than to shift a metal's.

More than a simple average

A plain average of the two temperatures is correct only when both heat capacities match — equal mass and equal specific heat. Otherwise the substance with more mass times specific heat resists change harder and drags the result toward itself. Dropping a hot iron bolt into a bucket of water barely warms the water, because the water's heat capacity dwarfs the bolt's. The model here assumes no boiling or freezing; if a substance changes state, its latent heat must be accounted for separately.