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Mixing Temperature Calculator
Computes the equilibrium temperature reached when two or three bodies of known mass, specific heat and initial temperature exchange heat, from energy conservation Σ m·c·(T − T_final) = 0, and the heat that flows from the hotter to the cooler bodies.
When to use
You mix hot and cold water, drop a heated metal into a liquid, or combine up to three substances and need the final common temperature or the heat exchanged (calorimetry without phase change).
Do not use when: A substance melts, boils or freezes during the exchange (latent heat is not included), the container or surroundings absorb significant heat, or you only need the heat for a given temperature change of one body (use specific-heat).
Formula
final_temperature_c = (m1·c1·T1 + m2·c2·T2 + m3·c3·T3) / (m1·c1 + m2·c2 + m3·c3); heat_transferred_j = Σ over bodies with T_i > T_final of m_i·c_i·(T_i − T_final) (= m1·c1·|T1 − T_final| for two bodies); °F = °C × 9/5 + 32; K = °C + 273.15
Energy conservation in an ideal calorimeter: heat lost by the hotter bodies equals heat gained by the cooler ones, with constant specific heats, no phase change and no heat exchange with the container or surroundings.
Inputs
| Parameter | Type | Unit | Required | Description |
|---|---|---|---|---|
mass_1_kg | number | kg | yes | Mass of the first body in kilograms (1 L of water ≈ 1 kg). Range: > 0 |
specific_heat_1_j_kg_k | number | J/(kg·K) | default 4186 | Specific heat capacity of body 1: water 4186, ice 2090, aluminium 897, iron 449, copper 385, ethanol 2440. Range: > 0, ≤ 100000 |
temperature_1_c | number | °C | yes | Initial temperature of body 1 in °C. Range: ≥ -273.15, ≤ 5000 |
mass_2_kg | number | kg | yes | Mass of the second body in kilograms. Range: > 0 |
specific_heat_2_j_kg_k | number | J/(kg·K) | default 4186 | Specific heat capacity of body 2 (default: liquid water, 4186). Range: > 0, ≤ 100000 |
temperature_2_c | number | °C | yes | Initial temperature of body 2 in °C. Range: ≥ -273.15, ≤ 5000 |
mass_3_kg | number | kg | default 0 | Mass of an optional third body in kilograms; leave at 0 for a two-body mixture. Range: ≥ 0 |
specific_heat_3_j_kg_k | number | J/(kg·K) | default 4186 | Specific heat capacity of body 3 (default: liquid water, 4186). Range: > 0, ≤ 100000 |
temperature_3_c | number | °C | no | Initial temperature of body 3 in °C; required when mass_3_kg is greater than 0. Range: ≥ -273.15, ≤ 5000 |
Outputs
| Output | Type | Unit | Description |
|---|---|---|---|
final_temperature_c | number | °C | Common equilibrium temperature T_final = Σ m·c·T / Σ m·c. |
final_temperature_f | number | °F | Equilibrium temperature in degrees Fahrenheit. |
final_temperature_k | number | K | Equilibrium temperature in kelvin. |
heat_transferred_j | number | J | Heat given up by the bodies hotter than T_final (equal to the heat absorbed by the cooler ones); for two bodies m1·c1·|T1 − T_final|. |
heat_transferred_kj | number | kJ | Heat transferred in kilojoules. |
note | string | Which bodies cool and which warm, and the assumptions (constant specific heats, no phase change, no losses). |
Example
1 kg water at 80 °C + 2 kg water at 20 °C: {"mass_1_kg":1,"temperature_1_c":80,"mass_2_kg":2,"temperature_2_c":20} → {"final_temperature_c":40,"final_temperature_f":104,"final_temperature_k":313.15,"heat_transferred_j":167440,"heat_transferred_kj":167.44}
0.5 kg copper (385 J/kg·K) at 100 °C into 1 kg water at 20 °C: {"mass_1_kg":0.5,"specific_heat_1_j_kg_k":385,"temperature_1_c":100,"mass_2_kg":1,"temperature_2_c":20} → {"final_temperature_c":23.52,"heat_transferred_j":14722.94,"heat_transferred_kj":14.7229}
GET https://tttkmbb.com/api/v1/calculate/mixing-temperature?mass_1_kg=1&temperature_1_c=80&mass_2_kg=2&temperature_2_c=20
Machine access
- API:
GET https://tttkmbb.com/api/v1/calculate/mixing-temperature(query parameters) orPOSTwith a JSON body{"inputs": {...}} - Schema: https://tttkmbb.com/api/v1/calculators/mixing-temperature · Markdown: https://tttkmbb.com/physics/mixing-temperature.md · JSON definition: https://tttkmbb.com/physics/mixing-temperature.json
- MCP: server
https://tttkmbb.com/mcp, toolrun_calculator with calculator_id="mixing-temperature" - OpenAPI operationId:
calculate_mixing_temperature - Freshness:
static. Authentication: none. Rate limit: fair use (see rate limits).
Sources
- Wikipedia – Calorimetry (reference)
- Wikipedia – Specific heat capacity (reference)
- HyperPhysics – Specific heat (reference)
FAQ
Why is the final temperature of copper in water so close to the water temperature?
Water's specific heat (4186 J/kg·K) is about 11 times copper's (385), so 1 kg of water has far more heat capacity than 0.5 kg of copper and its temperature barely moves.
Can I mix ice and water with this calculator?
Only if no ice melts. Melting takes 334 kJ/kg of latent heat that this calculator ignores, so for ice-water mixtures the real final temperature is lower than the result.
Does the container matter?
A real container absorbs some heat. Enter it as body 3 (mass × specific heat of the vessel, at the initial temperature of its contents) to include it.
Related calculators
- Specific Heat Calculator — Heat for a given temperature change of a single body (Q = m·c·ΔT).
- Temperature Converter — Convert temperatures between °C, °F and K.
- Thermal Expansion Calculator — Size change of a solid caused by the temperature change.