Thermal Equilibrium Calculator — Final Mixed Temperature
Calculate the final temperature when two objects at different temperatures exchange heat until they reach equilibrium. Enter the mass, specific heat, and initial temperature of each object to find Tf and the heat transferred.
kg
J/(kg·K)
°C
kg
J/(kg·K)
°C
Temperature both objects reach when heat exchange is complete
- 1
Heat capacity A: m₁ × c₁
0.5 × 4,186 = 2,093 - 2
Heat capacity B: m₂ × c₂
1 × 4,186 = 4,186 - 3
Numerator: m₁c₁T₁ + m₂c₂T₂
2,093 × 80 + 4,186 × 20 = 251,160 - 4
Equilibrium temperature Tf
251,160 ÷ (2,093 + 4,186) = 40Weighted average — the object with the larger heat capacity pulls Tf toward its initial temperature.
How does this calculator work?
When two objects exchange heat in an insulated system, the final equilibrium temperature is Tf = (m₁c₁T₁ + m₂c₂T₂)/(m₁c₁ + m₂c₂) — a weighted average skewed toward the object with the larger heat capacity mc. Enter mass, specific heat, and initial temperature for each object.
Formula
How this is calculated
When two objects at different temperatures are placed in contact in an insulated system, heat flows from the hotter to the cooler until both reach the same final temperature Tf. By the conservation of energy (heat lost by the hot object equals heat gained by the cool one), the equilibrium condition is m₁c₁(T₁ − Tf) = m₂c₂(Tf − T₂). Solving for Tf gives the weighted average formula Tf = (m₁c₁T₁ + m₂c₂T₂)/(m₁c₁ + m₂c₂), where m is mass, c is specific heat capacity, and the product mc is the total heat capacity of each object.
The final temperature always lies between the two initial temperatures. An object with a larger heat capacity (large mc) pulls Tf closer to its own starting temperature — this is why adding a small amount of hot metal to a large bath of cold water barely changes the water temperature.
This calculation assumes a perfectly insulated (adiabatic) container with no heat loss to the surroundings and no phase changes. If either substance melts, boils, or freezes during the process, additional latent heat terms must be included.
Frequently asked questions
The final temperature is a weighted average where the weight is each object's heat capacity mc. A larger or denser object (higher mc) stores more thermal energy, so it shifts Tf further toward its own initial temperature. Only if both objects have identical heat capacities does Tf equal the simple arithmetic mean.
The formula assumes a perfectly insulated (adiabatic) system — all heat lost by the hotter object is gained by the cooler one. In reality, some heat escapes to the environment. For quick experiments this error is small; for precise calorimetry a correction factor or an insulated calorimeter is used.
Yes — extend the formula: Tf = (Σ mᵢcᵢTᵢ) / (Σ mᵢcᵢ). Add as many (m, c, T) terms as needed in the numerator and denominator. This calculator handles two objects; run it in sequence for more.
Also known as
TG we-Calculate Editorial Team. (2026). Thermal Equilibrium Calculator — Final Mixed Temperature [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/thermal-equilibrium-calculator
TG we-Calculate Editorial Team. "Thermal Equilibrium Calculator — Final Mixed Temperature." TG we-Calculate. 2026. https://we-calculate.com/calculator/thermal-equilibrium-calculator.
TG we-Calculate Editorial Team, "Thermal Equilibrium Calculator — Final Mixed Temperature," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/thermal-equilibrium-calculator
@misc{wecalculate_thermal_equilibrium_calculator, title = {Thermal Equilibrium Calculator — Final Mixed Temperature}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/thermal-equilibrium-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?