Calculation
Tm = (m1·c1·T1 + m2·c2·T2) / (m1·c1 + m2·c2)
At equilibrium, heat given off by the warmer mass equals heat absorbed by the colder mass (the mixing rule).
Dubbel Thermodynamik D 6.4 Mischtemperatur: tm = Σ(mi·cpi·ti)/Σ(mi·cpi)
Mixing temperature of two masses
At equilibrium, heat given off by the warmer mass equals heat absorbed by the colder mass (the mixing rule).
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Mixing temperature Tm — Common equilibrium temperature after complete thermal equalisation.
- Mass 1 m1 — Mass of the first, usually warmer, quantity.
- Specific heat capacity c1 — Material property of the first mass; water about 4,187 J/(kg·K).
- Temperature 1 T1 — Initial temperature of the first mass before mixing.
- Mass 2 m2 — Mass of the second, usually colder, quantity.
- Specific heat capacity c2 — Material property of the second mass; water about 4,187 J/(kg·K).
- Temperature 2 T2 — Initial temperature of the second mass before mixing.
Example
2 kg of water at 80 °C and 3 kg of water at 20 °C give Tm=(2·80+3·20)/5=44 °C.
Assumptions and limits
No heat loss to surroundings or vessel, constant specific heat capacities and no phase change during mixing.
Understand Mixing temperature of two masses
This calculator determines the common equilibrium temperature reached when mixing two masses of different temperature and heat capacity, a classic calorimetry basic case.
What does this quantity describe?
At equilibrium the warmer mass releases as much heat as the colder one absorbs: m1·c1·(T1−Tm)=m2·c2·(Tm−T2). Solved for Tm, this gives the weighted average Tm=(m1·c1·T1+m2·c2·T2)/(m1·c1+m2·c2).
Formula and variables
Tm = (m1·c1·T1 + m2·c2·T2) / (m1·c1 + m2·c2)
Tm = (m1·c1·T1 + m2·c2·T2) / (m1·c1 + m2·c2)
| Symbol / input | Meaning |
|---|---|
| Mixing temperature Tm | Common equilibrium temperature after complete thermal equalisation. |
| Mass 1 m1 | Mass of the first, usually warmer, quantity. |
| Specific heat capacity c1 | Material property of the first mass; water about 4,187 J/(kg·K). |
| Temperature 1 T1 | Initial temperature of the first mass before mixing. |
| Mass 2 m2 | Mass of the second, usually colder, quantity. |
| Specific heat capacity c2 | Material property of the second mass; water about 4,187 J/(kg·K). |
| Temperature 2 T2 | Initial temperature of the second mass before mixing. |
Choose the inputs correctly
m1, c1 and T1 are mass, specific heat capacity and initial temperature of the first quantity; m2, c2 and T2 the corresponding values for the second.
How to use the calculator
Weigh masses or derive them from volume and density; take specific heat capacities from material tables (water: 4,187 J/(kg·K)).
Worked example
2 kg of water at 80 °C and 3 kg of water at 20 °C give Tm=(2·80+3·20)/5=44 °C.
Understand the result and units
The larger mass or larger heat capacity dominates the mixing temperature; for the same substance with very different masses, Tm sits closer to the larger quantity's temperature.
m1 and m2 are masses, c1 and c2 specific heat capacities, T1, T2 and Tm temperatures.
Typical applications
Estimating mixing temperatures when combining two liquid streams, in calorimetry, and for thermal-storage charging processes.
Assumptions, limits and common mistakes
No heat loss to surroundings or vessel, constant specific heat capacities over the temperature range, and no phase change (e.g. freezing, evaporation) during mixing.
Common mistake: Do not confuse specific heat capacity with heat capacity (m·c); the formula needs both masses and both specific values separately.
Frequently asked questions
What is “Mixing temperature of two masses” used for?
Estimating mixing temperatures when combining two liquid streams, in calorimetry, and for thermal-storage charging processes.
Where do the input values come from?
m1, c1 and T1 are mass, specific heat capacity and initial temperature of the first quantity; m2, c2 and T2 the corresponding values for the second.
What does the result not cover?
No heat loss to surroundings or vessel, constant specific heat capacities over the temperature range, and no phase change (e.g. freezing, evaporation) during mixing.