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Enter values to see results
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°C
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kJ
The Thermal Equilibrium Calculator determines the final temperature when two objects at different temperatures are brought into thermal contact. Based on the Zeroth Law of Thermodynamics and conservation of energy, heat flows from the hotter object to the cooler one until both reach the same temperature.
The equilibrium temperature is found by setting the heat lost by the hot object equal to the heat gained by the cold object: $$m_1 c_1 (T_1 - T_f) = m_2 c_2 (T_f - T_2)$$ Solving for the final temperature: $$T_f = \frac{m_1 c_1 T_1 + m_2 c_2 T_2}{m_1 c_1 + m_2 c_2}$$
This is essentially a weighted average of the two initial temperatures, where the weights are the thermal masses (mass times specific heat capacity) of each object. An object with a large thermal mass has a greater influence on the final temperature—this is why a small amount of hot water quickly cools when added to a large cold bath.
Thermal equilibrium calculations are fundamental in calorimetry, cooking, industrial mixing, HVAC system design, and environmental science. When you mix hot and cold water for a bath, pour hot coffee into a cold mug, or design a heat exchanger, you are implicitly solving this equation. In laboratory calorimetry, measuring the equilibrium temperature allows determination of unknown specific heat capacities or heats of reaction.
Common specific heat values include: water at 4186 J/(kg·K), aluminum at 897 J/(kg·K), iron at 449 J/(kg·K), copper at 385 J/(kg·K), and glass at 840 J/(kg·K). Water's exceptionally high specific heat makes it an excellent coolant and thermal buffer, which is why coastal climates are more moderate than inland ones.
This calculator assumes no heat loss to the environment (a perfectly insulated system), no phase changes, and constant specific heat capacities over the temperature range. For more complex scenarios involving phase transitions, the latent heat must be included separately.
The thermal equilibrium calculator applies conservation of energy to a two-body system:
Energy Balance:
$$Q_{\text{lost}} = Q_{\text{gained}}$$
$$m_1 c_1 (T_1 - T_f) = m_2 c_2 (T_f - T_2)$$
Equilibrium Temperature:
$$T_f = \frac{m_1 c_1 T_1 + m_2 c_2 T_2}{m_1 c_1 + m_2 c_2}$$
This is a thermal-mass-weighted average. The quantity $$mc$$ (mass × specific heat) is called the heat capacity of the object—it determines how much energy is needed to change the object's temperature by one degree.
Heat Transferred:
$$Q = m_1 c_1 |T_1 - T_f| = m_2 c_2 |T_f - T_2|$$
Both expressions give the same result (conservation of energy). The calculator reports the absolute value of heat transferred between the two objects.
The equilibrium temperature always falls between the two initial temperatures, closer to the temperature of the object with the larger thermal mass (m×c). If both objects have the same thermal mass, the final temperature is the simple arithmetic mean. The heat transferred tells you how much energy moved from the hot object to the cold one. This assumes perfect insulation—in real situations, some heat escapes to the surroundings, and the actual equilibrium temperature may be slightly lower. For objects with very different specific heats (e.g., water vs. metal), the material with the higher specific heat dominates the final temperature.
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0.5 kg of water at 80°C is mixed with 1 kg of water at 20°C. Since both have the same specific heat, the final temperature is a mass-weighted average: (0.5×80 + 1×20)/1.5 = 40°C. 83.7 kJ of heat transfers from hot to cold water.
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A 100 g copper block at 200°C (c = 385 J/(kg·K)) is dropped into 500 g of water at 25°C. The equilibrium temperature is only 28.2°C because water's thermal mass (2093 J/K) far exceeds copper's (38.5 J/K). Only 6.6 kJ of heat transfers.
The final temperature is a weighted average where the weight of each object is its thermal mass (m × c). An object with large thermal mass can absorb or release a lot of heat with little temperature change, so it 'pulls' the final temperature toward its initial value. Water has a very high specific heat (4186 J/(kg·K)), so a large mass of water barely changes temperature when a small hot metal is dropped in.
Specific heat capacity (c) is the amount of energy required to raise the temperature of 1 kg of a substance by 1 K (or 1°C). It measures a material's ability to store thermal energy. Water has an unusually high specific heat (4186 J/(kg·K)), metals have low values (copper: 385, iron: 449), and air has about 1005 J/(kg·K). Higher specific heat means the material resists temperature changes and can store more thermal energy per unit mass.
No. This calculator assumes no phase transitions (melting, boiling, freezing) occur. If a phase change happens—for example, ice melting in warm water—the latent heat of the phase transition must be included. During a phase change, temperature remains constant while heat is absorbed or released. For problems involving phase changes, you would need to add the latent heat term: Q = mL, where L is the latent heat of fusion or vaporization.
The principle extends directly to any number of objects. The general formula is: T_f = Σ(m_i c_i T_i) / Σ(m_i c_i). For two objects, this gives the formula in this calculator. For three or more objects, simply add all the m×c×T products in the numerator and all the m×c products in the denominator. You can use this calculator iteratively: find equilibrium for objects 1 and 2, then use that result as one object and add object 3.
The calculator assumes: (1) a perfectly insulated system with no heat loss to surroundings, (2) no phase changes occur, (3) specific heat capacities are constant over the temperature range, (4) the objects reach true equilibrium (uniform final temperature), and (5) no work is done on or by the system. In practice, heat losses reduce the final temperature, and c may vary with temperature, but the errors are usually small for moderate temperature ranges.
Calorimetry is the experimental measurement of heat transfer, and thermal equilibrium is its core principle. In a simple calorimeter, a hot object is placed in water of known mass and temperature. By measuring the equilibrium temperature, you can calculate the heat lost by the object and determine its specific heat capacity: c = Q/(mΔT). This technique is used in chemistry to measure heats of reaction, in materials science to characterize new materials, and in food science to determine caloric content.
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