Newton’s Law of Cooling Calculator

A hot object cools fastest when it is hottest — its temperature decays exponentially toward its surroundings’. Enter the starting and ambient temperatures, the cooling constant and the elapsed time, and this calculator returns the temperature now.

The Exponential Approach to Room Temperature

Newton stated the law in 1701 (anonymously, in Latin, in the Philosophical Transactions): the rate of cooling is proportional to the temperature excess, dT ÷ dt = −k(T − T_s). Integrated, the excess decays by the same fraction in each equal time — T(t) = T_s + (T₀ − T_s)e^−kt — the signature exponential of a hundred relaxation processes (capacitor discharge, radioactive decay's mirror image, drug elimination). The constant k collects everything practical — surface area, insulation, draughts, the object's heat capacity — into one number with units of "per time," measurable from two thermometer readings: k = ln(excess₁ ÷ excess₂) ÷ Δt. The calculator runs the integrated curve; the half-excess time it prints (ln 2 ÷ k) is the intuitive handle: every that-many minutes, the gap to room temperature halves, forever approaching, never (in the ideal law) arriving.

Worked Example: Coffee, Ten Minutes On

The defaults are the desktop tragedy: coffee poured at 90 °C into a 20 °C room, k = 0.05 per minute. After 10 minutes: T = 20 + 70 × e^−0.5 = 20 + 70 × 0.6065 = 62.5 °C — the excess (70°) fallen to 42.5°, 60.7% of its start. The half-excess time is 13.9 minutes, so at 24 minutes the coffee is 41 °C, at 38 minutes 30.5 °C — drinkable decisions all along the curve. Note the shape's mercy and cruelty: the first minutes are the expensive ones (the coffee loses a quarter of its excess before you have answered two emails), while the last degrees linger asymptotically — the law that food-safety charts (cool the soup through the danger zone fast) and forensic pathologists both read professionally.

Algor Mortis: The Law at the Inquest

The cooling curve's most solemn employment is estimating time of death: a body cools from 37 °C toward ambient along this exponential (with a plateau in the first hour as internal gradients settle), and the forensic textbooks (Henssge's nomogram refines the single-k model with body mass and clothing corrections) invert the calculator — temperature and ambient in, hours since death out. Newton's law, stated for a cooling iron, became murder investigations' quiet instrument, and its error bars are argued in court. The calculator's forward direction is the same equation the nomogram inverts; the assumptions listed in its note (uniform temperature, constant ambient, constant k) are precisely the ones the forensic refinements exist to repair.

The Same Curve in the Capacitor

Engineers meet this exponential so often they give its constant a Greek letter and a unit of honour: an RC circuit's voltage decays as e^{−t ÷ RC}, a damped spring's amplitude likewise, a drug's plasma level likewise — k and 1 ÷ RC are the same idea, the rate of a system forgetting its displacement from equilibrium. Newton's cooling is the thermal member of the family, and its lesson transfers verbatim: measure any two points, take the logarithm ratio, and the whole future of the relaxation is known. The calculator computes the family's founding case; the reader who has used it once will recognise the curve in a dozen other subjects' clothing, which is what a physical law is for.

Building Physics: The Night Purge

Architects run this curve on whole buildings: a heavyweight concrete frame, heated through a summer day, cools toward the night air along an exponential whose k the ventilation rate sets — open the vents at night (the "night purge") and k rises, flushing the day's stored heat so the frame starts the next morning cool, a passive air-conditioning cycle computed in exactly this page's terms at tonne scale. The thermal mass that the capacity page prices is the numerator; the purge's airflow is the denominator's lever. Buildings that ignore the curve overheat on schedule; buildings designed around it breathe on schedule instead — the difference is one constant, honestly measured.

Food Safety's Danger Zone

Public-health codes are this curve with legal force: cooked food must pass from 60 °C down through the bacterial banquet band (roughly 5–60 °C) within set hours, which is a requirement on the soup's k — achieved by shallow pans (more surface, larger k), ice baths, and refrigeration that holds the ambient term low. The exponential explains the rules' shape: cooling is fastest at the start, so the first hour does most of the work, and a deep stockpot's centre — where the uniform-temperature assumption fails worst — is the inspector's legitimate worry. The calculator models the ideal pot; the regulations exist for the real ones.

Related Tools

More from the Physics Calculators hub: the Heat Capacity Calculator — the capacity half of what sets k — and the Latent Heat Calculator, for the cooling that pauses at a phase change instead of following the curve.

Frequently Asked Questions

How do I find k for my own object?

Measure twice: record the temperature at two known times, compute the excesses over ambient, and take k = ln(excess₁ ÷ excess₂) ÷ (t₂ − t₁). Two thermometer readings and a logarithm — the calculator then extrapolates the whole curve from your pair. A mug of coffee in still air gives 0.03–0.06 per minute; a lidded, preheated mug much less; a steel casting in a foundry draught much more. The constant is a property of the situation (object + surroundings + airflow), not the substance alone, which is why the calculator asks for it instead of pretending a table exists.

Does the law work for warming too?

Identically — a cold object in warm surroundings warms along the same exponential approach, excess negative and shrinking; frozen pipes thawing, a roast warming toward oven temperature from below, and the Earth itself responding to a changed climate forcing all follow the curve with the sign flipped. The equation is about relaxation toward equilibrium, not about cold; Newton's name for it stuck to the cooling direction because his experiments (a red-hot iron in wind) ran that way, but the mathematics is even-handed, and so is the calculator.

Why does real cooling deviate — evaporation and radiation?

Because k bundles every loss channel and not all are linear in the excess: radiation grows as T⁴ (the Stefan–Boltzmann law — at furnace temperatures it swamps the linear term, which is why Newton's experiments at high heat needed his later corrections), and evaporation adds a latent-heat leak that depends on humidity. For the range of a coffee mug, a casting on the bench, a body in a room — tens of degrees of excess — the linear law fits within measurement error, and the calculator's domain statement is the honest one: small excesses, steady surroundings, stirred or thin objects. Beyond that, k becomes a function of T and the exponential needs numerical company.

What is the time constant?

τ = 1 ÷ k — the time for the excess to fall to 1/e (36.8%) of itself, and the more natural unit of the curve for engineers (thermal, electrical RC, and mechanical damping all share the dialect). For the worked example, τ = 20 minutes: after 20 minutes the coffee's excess is 25.7° of its original 70. After three time constants, 5% remains — "effectively equilibrated" by most benches' standards. The widget prints the halving time instead (ln 2 × τ) because halves are friendlier than e's, but the two are one multiplication apart and the calculator's k serves both.

Can the formula say when the coffee was poured?

Invert it: t = ln((T₀ − T_s) ÷ (T − T_s)) ÷ k — if you trust the starting temperature (fresh coffee's 90–95 °C is a decent prior) and have measured k for the mug from a previous experiment, a thermometer reading now timestamps the pour. It is the forensic inversion in miniature, complete with its weakness: every assumption (pour temperature, steady room, untouched mug) smears the estimate, which is why detectives quote ranges and why your own use — "has this been sitting here an hour?" — should quote one too. The calculator's forward arithmetic, run at a few trial times, brackets the answer honestly.

Does stirring or wind change the answer?

It changes k, not the law's shape: moving fluid strips the insulating boundary layer from the surface, raising the heat-transfer coefficient and with it k — Newton's own experiments were done in wind for this reason, and the foundry's draught, the laptop's fan and the swimmer's chill in a current are all k being revised upward by motion. In practice, doubling the airflow does not quite double k (boundary-layer physics gives a sub-linear law), but the calculator's contract is unchanged: measure k in the conditions you care about, and the exponential does the rest.

Who actually stated this law — and when?

Newton, in a 1701 paper on a temperature scale of his own devising ("Scala graduum Caloris"), published anonymously as was his habit with unfinished business. The proportionality he reported came from experiments with a glowing iron block cooling in wind — the forced-convection regime where the law fits best — and the qualification his modern readers add (gentle conditions, moderate excesses) is the same one this page posts. Three centuries on, the law remains the first model reached for in every cooling problem, which is the definition of a classic: not the final word, but the first, and usually enough.