Thermal engineering · Thermodynamics

Thermal equilibrium temperature calculator

Find the common final temperature of two isolated bodies with known masses, specific heats and initial temperatures. Signed heat values show which body gains or loses energy.

Created Oct 3, 2026v1.0

By Thibaut Grzelak, Mechanical Analysis Engineer

Thermodynamics

Two bodies exchange heat from the hotter to the colder until both reach Tf. Equal temperatures produce no heat-transfer arrow.

Quantity to calculate

Mass 1

Strictly positive body mass.

Specific heat 1

Positive constant specific heat, without phase change.

Temperature 1

Initial temperature ≥ 0 K; select the unit explicitly.

Mass 2

Strictly positive body mass.

Specific heat 2

Positive constant specific heat, without phase change.

Temperature 2

Initial temperature ≥ 0 K; select the unit explicitly.

Results

Equilibrium temperature

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Final common temperature in an isolated two-body system.

Heat received by body 1

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Negative if body 1 cools.

Heat received by body 2

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Opposite of Q₁, by energy conservation.

Relation and numerical substitutionT_f = (m1 × c1 × T1 + m2 × c2 × T2) / (m1 × c1 + m2 × c2)

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Two-body isolated system, constant positive specific heats, no phase change or chemical reaction, negligible container heat capacity.

01Formulas and symbols

Formulas used

T_fT_f = (m1 × c1 × T1 + m2 × c2 × T2) / (m1 × c1 + m2 × c2)Final common temperature in an isolated two-body system.

Q₁Q₁ = m1 × c1 × (T_f − T1)Negative if body 1 cools.

Q₂Q₂ = m2 × c2 × (T_f − T2)Opposite of Q₁, by energy conservation.

02Assumptions and limits

Scope of validity

Two-body isolated system, constant positive specific heats, no phase change or chemical reaction, negligible container heat capacity.

03Validation example

Reference numerical case

Two 1 kg bodies with c = 4186 J/(kg·K), initially at 60 °C and 20 °C, reach 40 °C. Q₁ = −83720 J and Q₂ = +83720 J; Q₁ + Q₂ = 0.

04References

Technical references

University Physics 2, §1.4 Heat transfer, specific heat and calorimetry — OpenStax

FAQ

Must the final temperature lie between the initial temperatures?

Yes, for positive masses and specific heats in this isolated, no-phase-change model.

What do negative heat values mean?

A negative Qi means body i loses heat. The other body receives that energy; their sum is zero up to numerical round-off.

05Model and conventions

Find the common final temperature of two isolated bodies with known masses, specific heats and initial temperatures. Signed heat values show which body gains or loses energy.

06Limits of this model

A phase transition invalidates this formula. Heat losses and the vessel may shift the real final temperature; this tool does not predict the time required to equilibrate.

07Common mistakes

Weight temperatures by mc, not mass alone unless specific heats are equal. Celsius and Fahrenheit are converted to kelvins before the energy balance.