When hot and cool water mix in an ideal insulated container, the energy lost by the hot water equals the energy gained by the cool water. Both end at the same temperature.
Use energy transferred = mass × specific heat capacity × temperature change. Since both samples are water, the same specific heat capacity appears on both sides and cancels.
- Call the final temperature T.
- Use initial hot temperature minus T for the fall in temperature.
- Use T minus initial cool temperature for the rise.
Worked example | Mix 150 g and 250 g of water
Mix 150 g of water at 80°C with 250 g at 24°C. Assume no energy is transferred to the container or surroundings. The hot water cools by 80 − T degrees and the cool water warms by T − 24 degrees.
Cancelling the specific heat capacity gives 150(80 − T) = 250(T − 24). Expand to get 12,000 − 150T = 250T − 6,000. Then 18,000 = 400T, so the final temperature is 45°C.
Check which starting temperature it is closer to
The result is between 24°C and 80°C and closer to 24°C because the cooler portion has more mass. At 45°C, the hot water falls by 35°C and the cool water rises by 21°C. The products 150 × 35 and 250 × 21 are both 5,250 g °C, so multiplying each by the same specific heat capacity gives equal energy changes.
Try a larger difference in mass
Mix 100 g of water at 70°C with 300 g at 30°C under the same assumptions. Find the final temperature.
Answer. Solve 100(70 − T) = 300(T − 30) to get 40°C. The heavier portion warms by 10°C while the lighter portion cools by 30°C.
Frequently asked questions
When can I use the simple average of the two temperatures?
For two equal masses of the same substance under the stated assumptions, the two temperature changes have equal magnitudes. The final temperature is then the midpoint.
What changes if the liquids have different specific heat capacities?
Keep each liquid’s specific heat capacity in its own mass × specific heat capacity × temperature change expression.
Official information
SEAB, 2027 SEC G3 Physics syllabus (K323), Thermal Properties of Matter, specific heat capacity. Syllabus checked 11 November 2026. The examples are original.