Newton's Law of Cooling Calculator
Newton's Law of Cooling (Newton 1701) says an object's temperature approaches its surroundings exponentially: T(t) = T∞ + (T₀ − T∞)·e^(−kt). Enter the initial temperature, ambient temperature, cooling constant k and elapsed time to get the current temperature, remaining gap, percent cooled, plus the half-cooling time τ½ = ln 2 / k and time constant τ = 1/k. An optional target temperature is solved for the time needed to reach it. Units are unconstrained as long as T₀ and T∞ share one (°C, °F or K) and k uses the inverse of t's time unit.
Enter a valid initial and ambient temperature, a positive cooling constant and a non-negative time.
Temperature T(t)
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Related figures
- Remaining gap (T − T∞)
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- Percent cooled
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- Half-cooling time τ½ = ln 2 / k
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- Time constant τ = 1 / k
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Time to reach target —
Only valid when the target lies between the initial and ambient temperatures (i.e. on the way to ambient).
Formula
T(t) = T∞ + (T₀ − T∞) · e^(−k · t)
Formula
T(t) = T∞ + (T₀ − T∞) · e^(−k · t) τ½ = ln 2 / k τ = 1 / k t_to_target = (1/k) · ln((T₀ − T∞)/(T_target − T∞))
- · The cooling constant k depends on geometry, surface area, density, specific heat and the convection/radiation environment — it is not a material property and must be measured for each scenario (e.g. coffee mug k ≈ 0.03–0.06 /min, insulated bottle k ≈ 0.01–0.02 /min).
- · Easiest way to estimate k: record temperature at two times T₁ and T₂, then k = ln((T₁ − T∞)/(T₂ − T∞)) / Δt.
- · Half-cooling time τ½ = ln 2 / k ≈ 0.693 / k: the temperature gap halves every τ½. After 3·τ½ the gap is ~12.5 % of its original value; after 5·τ½ it is ~3 %.
- · Time constant τ = 1/k: the elapsed time for the gap to fall to 1/e ≈ 36.8 % of its initial value — the standard engineering characteristic timescale.
- · The law is a "lumped capacitance" approximation: it assumes uniform internal temperature, a constant convection coefficient and negligible radiation. Accuracy degrades for very large temperature differences (e.g. above ~200 °C where radiation matters) or for large, low-conductivity objects.
- · The forensic Glaister equation (time of death ≈ (37 − T_rectal) / 1.5 hours) is a linearised Newton-cooling rule. Modern forensic practice uses Henssge's nomogram and similar models that account for body mass, clothing and environment.
- · References: Incropera et al., Fundamentals of Heat and Mass Transfer, ch. 5 (lumped capacitance); Holman, Heat Transfer, ch. 4.
Frequently asked
How is the cooling constant k determined? Are there typical values?
k is not a material constant — it depends on geometry, surface area, density, specific heat and the convective environment, so each setup must be measured. Rough typical values (units /min): hot coffee in a ceramic mug ≈ 0.03–0.06; hot soup in a metal pot ≈ 0.05–0.10; water bottle at room temperature ≈ 0.01–0.02; insulated travel mug ≈ 0.002–0.005. To measure your own k, log the temperature at two times and solve k = ln((T₁ − T∞)/(T₂ − T∞)) / Δt with your ambient temperature T∞.
When can I drink my coffee? Brewed at 90 °C in a 22 °C room — when will it reach 60 °C?
Plug T₀ = 90, T∞ = 22, k ≈ 0.05 /min and target 60 °C into the tool: t = (1/0.05)·ln((90−22)/(60−22)) = 20·ln(68/38) ≈ 11.6 min. So about 12 minutes. The exact number depends on your mug shape, surface area and whether it has a lid — measure your own k once and you can predict future cups accurately.
What is the difference between the half-cooling time τ½ and the time constant τ?
Both describe the speed of exponential decay but at different thresholds. τ½ = ln 2 / k ≈ 0.693/k is the time for the gap to fall to half (50 %) — directly analogous to a radioactive half-life and intuitive. τ = 1/k is the time for the gap to fall to 1/e ≈ 36.8 %, the standard characteristic timescale used across engineering and physics. τ ≈ 1.443·τ½ for any exponential decay.
When does Newton's Law of Cooling break down?
In three main situations: (1) very large temperature differences (e.g. molten metal, furnace interiors), where radiation following the Stefan-Boltzmann T⁴ law dominates and the full conduction equation is needed; (2) large or low-conductivity objects (thick rock, big timber) where internal temperature is no longer uniform — the "lumped" assumption fails when the Biot number exceeds about 0.1, and a PDE is required; (3) changing surroundings (wind picks up, object moves) so that k itself drifts — solve piecewise or use a time-varying model. For everyday objects near room temperature, Newton's Law is usually accurate enough.
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