Find the final equilibrium temperature when bodies exchange heat: mix up to 5 bodies, melt ice in warm water (phase change included), or model two bodies in thermal contact with the time it takes to settle. Everything runs in your browser, no server, no sign-up.
Final temperature of an isolated mixture: Tf = Σ(m·c·T) / Σ(m·c). Pick a material preset or enter a custom specific heat.
Ice at or below 0 °C mixed with water at or above 0 °C. The solver balances warming the ice, melting it (latent heat 334,000 J/kg), and cooling the water, and reports whether all ice melts, part melts, or water freezes.
Two bodies pressed together exchange heat through a contact conductance G. The calculator finds the final equilibrium temperature, the thermal time constant τ, and how long it takes to settle within your target.
In an isolated system the heat lost by warm bodies equals the heat gained by cool bodies:
Each body's exchanged heat is qi = mi·ci·(Tf − Ti); the sum is zero by energy conservation.
Balance enthalpy against liquid water at 0 °C. Warming the ice costs mice·cice·(0 − Tice), melting it all costs mice·334,000 J/kg, and the water releases mwater·4,186·Twater. If the released heat covers warming plus melting, everything ends as water above 0 °C; otherwise the mixture settles at 0 °C with part of the ice unmelted (or part of the water frozen).
With thermal capacities C1 = m1c1, C2 = m2c2 and contact conductance G:
Time to settle within a target difference: t = τ·ln(ΔT0 / ΔTtarget).
| Material | J/(kg·K) | BTU/(lb·°F) |
|---|
Thermal equilibrium is the state where two or more bodies in thermal contact have reached the same temperature, so no net heat flows between them. For an isolated mixture with no phase change, the final temperature is the heat-capacity-weighted average Tf = (m1*c1*T1 + m2*c2*T2 + ...) / (m1*c1 + m2*c2 + ...).
Multiply each mass by its specific heat and temperature, add those products up, then divide by the sum of the mass-times-specific-heat terms. Example: 100 g of aluminum at 90 °C mixed with 200 g of water at 20 °C gives Tf = 24,817 / 926.9, which is about 26.77 °C (80.19 °F). Use the Mixing tab above to do this for up to 5 bodies.
Three outcomes are possible. If the warm water holds enough heat to warm the ice to 0 °C and melt it completely (latent heat 334,000 J/kg), everything ends as water above 0 °C. Otherwise the mixture settles at 0 °C with some ice left unmelted, or with some water frozen. Example: 50 g of ice at -10 °C in 300 g of water at 25 °C. The water can release 31,395 J cooling to 0 °C; warming the ice costs 1,045 J and melting it all costs 16,700 J. Since 31,395 - 1,045 - 16,700 = 13,650 J remains, all the ice melts and the final temperature is about 9.32 °C.
For two bodies with thermal capacities C1 = m1*c1 and C2 = m2*c2 joined by a contact conductance G, the temperature difference decays exponentially with time constant T = C1*C2 / (G*(C1+C2)). The time to settle within a target difference is t = T * ln(start difference / target difference). Example: two 1 kg bodies of water at 90 °C and 20 °C with G = 10 W/K give T = 209.3 s, and reaching within 1 °C takes about 889 s, roughly 14.8 minutes. See the Contact tab for the full approach table.
Mass: grams, kilograms, ounces, and pounds. Specific heat: J/(kg·K), J/(g·°C), and BTU/(lb·°F). Temperature: °C, °F, and K. Energy results: J, kJ, BTU, and kcal. Contact conductance: W/K, kW/K, and BTU/(hr·°F).
No. Newton's law of cooling models a single body cooling against a fixed-temperature environment. The Contact tab here models two bodies warming and cooling each other, which is a different physical situation with its own time constant. For the single-body case, use our Newton's Law of Cooling Calculator. Looking for the Indonesian version of equilibrium calculations? See our Kalkulator Kalorimetri.
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