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Newton's Law of Cooling

Find temperature after time t with exponential cooling model

Parameters

°Cⓘ
°Cⓘ
s⁻¹ⓘ
sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Temperature T(t):
23.49;°C23.49;°C
Time Constant τ:
20.00;s20.00;s
Excess Remaining:
4.98;4.98;%

Examples

Coffee cooling

T₀=90°C, T_env=20°C, k=0.05, t=60 s.

    Short time

    t=10 s same params.

      Visualization

      Newton's Law of Cooling — Exponential Approach to Ambient Temperature

      Newton's law of cooling (1701) states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient (environment) temperature: dT/dt = −k(T − T_env), where k > 0 is the cooling constant (s⁻¹) depending on surface area, heat transfer coefficient, and thermal properties.

      The solution is exponential: T(t) = T_env + (T₀ − T_env)e^(−kt), where T₀ is initial temperature. The excess temperature (T − T_env) decays exponentially from (T₀ − T_env) toward zero. As t → ∞, T → T_env regardless of T₀.

      The time constant τ = 1/k is the time for the excess temperature to fall to 1/e ≈ 36.8% of its initial value. After τ, (T − T_env) = (T₀ − T_env)/e. Half-life for excess temperature: t_(1/2) = τ ln 2 ≈ 0.693τ.

      Valid when: (1) temperature difference is moderate (not so large that radiation dominates nonlinearly), (2) convection is the main heat transfer mode, (3) T_env is constant. For hot objects (>200 °C), radiative cooling (∝ T⁴) may require a modified model.

      If T₀ < T_env (cold object in warm room), the same equation describes heating: temperature rises exponentially toward T_env. Forensic science uses cooling curves to estimate time of death; food safety uses cooling time to limit bacterial growth.

      To find k experimentally, plot ln(T − T_env) vs t — the slope is −k. Larger surface area, better convection (fan), or lower heat capacity → larger k → faster cooling.

      Key Concepts

      • dT/dt = −k(T − T_env) — proportional cooling rate
      • T(t) = T_env + (T₀ − T_env)e^(−kt)
      • Time constant τ = 1/k; half-life = τ ln 2
      • Excess (T − T_env) decays exponentially
      • Works for heating when T₀ < T_env
      • Find k from slope of ln(T − T_env) vs t

      Real-World Applications

      • Coffee, tea, and food cooling time estimates
      • Forensic time-of-death analysis (body cooling)
      • HVAC transient room temperature response
      • Heat treatment and metallurgy cooling curves
      • Class 11 physics cooling curve experiments
      • Electronic component temperature decay after load removal

      Explore Further

      More thermodynamics tools

      Physics Equations

      Cooling Law:
      T(t)=Tenv+(T0−Tenv)e−ktT(t) = T_{env} + (T_0 - T_{env})e^{-kt}
      Time Constant:
      τ=1/k\tau = 1/k

      Step-by-Step Solution

      See how the main results are calculated.

      1

      Step 1: Cooling Model

      Equation:

      T(t)=Tenv+(T0−Tenv)e−ktT(t) = T_{env} + (T_0 - T_{env})e^{-kt}

      Explanation:

      Newton's law of cooling — rate ∝ temperature difference.

      2

      Step 2: Initial & Environment

      Result:

      T0=90,Tenv=20,k=0.05s−1,t=60sT₀ = 90, T_env = 20, k = 0.05 s⁻¹, t = 60 s

      Explanation:

      T₀ > T_env for cooling; k depends on surface area and convection.

      3

      Step 3: Exponential Factor

      Calculation:

      e−kt=e−3=0.049787e^{-kt} = e^{-3} = 0.049787

      Explanation:

      Factor decays from 1 toward 0.

      4

      Step 4: Temperature

      Calculation:

      T=20+(90−20)×0.049787=23.4851T = 20 + (90 - 20) \times 0.049787 = 23.4851

      Result:

      T=23.4851T = 23.4851
      5

      Step 5: Limit

      Long-time behavior.

      Explanation:

      As t → ∞, T → T_env.

      6

      Step 6: Half-Life Style

      Time constant τ = 1/k.

      Calculation:

      τ=1/k=20.00 s\tau = 1/k = 20.00 \text{ s}

      Explanation:

      After τ, excess temperature drops to ~37% of initial excess.

      Frequently Asked Questions (FAQ)

      Half-life vs τ?

      Half-life = τ ln(2) for excess temperature halving.

      Heating instead of cooling?

      Same form if T₀ < T_env — exponential approach upward.

      Find k from data?

      Plot ln(T−T_env) vs t — slope = −k.

      Why not linear?

      Driving force (T−T_env) shrinks as object cools.

      Radiation significant?

      At high T, radiation matters — Newton model may need extension.

      Practice MCQs

      1. As t → ∞, T approaches:
      2. Larger k means:
      3. Cooling curve is:
      4. If T₀ = T_env:
      5. τ = 1/k is:
      6. Newton cooling applies best when:

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