Temperature of a water mixture

Final temperature when mixing two amounts of water, or how much hot or cold water to add to reach a temperature.

What you want to calculate

Mass of the water you already have kg

1 L of water ≈ 1 kg.

Temperature of the water you already have °C

Mass of the water you add kg

1 L of water ≈ 1 kg.

Temperature of the water you add °C

Result

The heat given off by the hot water is gained by the cold water: m1 × T1 + m2 × T2 = (m1 + m2) × T, so T = (m1 × T1 + m2 × T2) ÷ (m1 + m2).

Example: 10 kg at 20 °C plus 5 kg at 80 °C. T = (10 × 20 + 5 × 80) ÷ 15 = 600 ÷ 15 = 40 °C.

To find how much water to add: m2 = m1 × (T − T1) ÷ (T2 − T).

Example: you have 10 L at 15 °C and want to reach 37 °C with water at 90 °C. m2 = 10 × (37 − 15) ÷ (90 − 37) = 220 ÷ 53 = 4.15 L; total 14.15 L.

It works the same for cooling: 2 L at 60 °C with water at 10 °C to bring it down to 40 °C: m2 = 2 × (40 − 60) ÷ (10 − 40) = 1.33 L.

It assumes no heat is lost: the cold container, the air and waiting time lower the real temperature by a few degrees. Add the hot water little by little and check with a thermometer.
Only valid for water or similar liquids. Milk (specific heat ≈ 3.9 kJ/(kg·K) versus 4.18 for water) gives an error of a few degrees; with flour, oil or metals the error is large.
In bread dough the final temperature also depends on the flour, the room and the kneading: use the result as a starting point.
For a baby's bath, always check with your elbow or a thermometer (≈ 37 °C) before putting the baby in. Above 50 °C water can scald in a few seconds.