JEE/NEET Physics · Thermal Properties of Matter series · Part 4 of 4 · All parts →
- Newton’s law of cooling: rate of cooling ∝ temperature difference with surroundings
- T(t) = T_s + (T₀ − T_s)e^(−kt) — exponential approach, never quite arriving
- Smaller ΔT → slower loss: cooling decelerates as it proceeds
- The whole series on one card: expansion, calorimetry, transfer, cooling
- Stefan’s law (radiation ∝ T⁴) and Wien’s law (peak λ ∝ 1/T) complete the picture
Hot tea cools fast at first, then ever slower — the bigger the temperature gap, the faster the leak. As the gap closes, so does the tap. That’s Newton’s law of cooling, and this finale card compresses the whole series. Part 4 of the Thermal Properties of Matter series.
- Newton’s law of cooling
- The exponential fade
- Radiation’s big guns: Stefan and Wien
- The master formula card
- Final practice set
- Recap
Newton’s Law of Cooling
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| T_s | surroundings’ temperature | °C |
| T₀ | object’s starting temperature | °C |
| k | cooling constant (surface, material, airflow) | per second |
The Exponential Fade
T(t) = T_s + (T₀ − T_s)e^(−kt): the excess decays exponentially — halving, halving again, approaching room temperature without a deadline. Tea scalds at 90°, sips at 60°, sits at 35°… forever chasing room temperature.
Radiation’s Big Guns: Stefan and Wien
Every body radiates power per area σT⁴ (Stefan; T in kelvin) — double the absolute temperature and radiation grows 16×. Hotter bodies also peak at shorter wavelengths (Wien: λ_max ∝ 1/T): that’s why iron glows red, then orange, then white — the visible diary of temperature.
The Master Formula Card
| What | Formula | Remember |
|---|---|---|
| Kelvin conversion | T(K) = T(°C) + 273 | ΔT same in both |
| Linear expansion | ΔL = αLΔT | 2α area, 3α volume |
| Sensible heat | Q = mcΔT | water c = 4186 |
| Latent heat | Q = mL | L_f 3.36×10⁵, L_v 2.26×10⁶ |
| Calorimetry | heat lost = heat gained | check melting budget |
| Conduction | Q/t = kAΔT/L | layers add resistance |
| Cooling | dT/dt = −k(T − T_s) | exponential fade |
| Stefan | P/A = σT⁴ | kelvins, fourth power |
| Wien | λ_max × T = constant | hotter → bluer |
Solved Examples
Rate ∝ ΔT: initial excess 60°, new excess 30° → 3 °C/min.
✔
Answer: 3 °C/min
P ∝ T⁴: (600/300)⁴ = 16×.
Halving the kelvin temperature cuts radiation 16-fold — absolute temperature rules. ✔
Answer: 16 : 1
Apply Newton’s law at the MEAN temperature of each interval: first interval mean 55°, excess 35; next interval (50→40) mean 45°, excess 25.
Rate ratio 25/35 → time = 5×35/25 = 7 min.
✔
Answer: ≈7 minutes
- Using Celsius in Stefan’s law. T⁴ demands kelvin — (600 K)⁴ ≠ (327 °C)⁴’s intent.
- Cooling proportional to T, not (T − T_s). The GAP drives cooling, not the object’s absolute temperature.
- Equal cooling per equal time. It’s exponential: fast early, slow late — never linear.
- Mean-temperature shortcut applied carelessly. Use each interval’s own mean excess — examiners set exactly this trap.
This Physics in Your Daily Life
- Hot drinks reach ‘perfect sipping’ in minutes but stay lukewarm for ages — the exponential fade of Newton’s law, experienced daily by every tea drinker.
- Thermal cameras and night vision — read your Stefan glow: bodies at 310 K radiate strongly in infrared that silicon eyes can see.
- Light bulb colours and star colours — red stars are cooler, blue-white hotter: Wien’s law across the whole sky.
- Oven mitts and double-walled cookware — engineered k, A, L stacks from the conduction law.
- Fever tracking on a chart — your 39 °C body cooling toward 37 °C follows roughly the same exponential approach medicine monitors.
A crowded theatre emptying through one door: at first, people pour out fast (huge pressure of numbers). As the theatre empties, fewer remain to push out — the outflow shrinks with the crowd. Hot objects empty their ‘excess temperature’ the same way: the gap itself is what drives the loss, so as the gap shrinks, so does the speed of shrinking.
Tea at 80° over 20° room: excess 60, cooling fast. At 40°: excess 20, cooling 3× slower. At 25°: excess 5, barely cooling. Same k throughout — only the gap changed. Half the excess is shed in equal characteristic times: the famous halving rhythm of exponentials.
Plot temperature against time: a steep dive flattening into a gentle glide that hugs the room-temperature line forever without touching it. Change the room’s temperature and the whole curve re-anchors to the new line — the picture explains iced drinks in summer vs winter.
Practice set (answers hidden — try first)
(NEET-level) Cooling rate ∝
(JEE Main-level) Doubling absolute temperature multiplies radiation by:
(NEET-level) Red-hot vs white-hot iron:
(Concept) A body cools fastest when:
(JEE Main-level) Excess halves every 5 min: from 60° excess, after 15 min the excess =
- rate ∝ (T − T_s)
- exponential approach, never arrival
- Stefan: σT⁴ (kelvin!)
- Wien: hotter peaks bluer
- mean-temperature method for intervals
- 🔁 Newton’s cooling law
- 🔁 exponential solution shape
- 🔁 Stefan and Wien basics
- 🧠 Chant: ‘the gap drives the leak’.
- 🧠 Stefan: ‘kelvins to the fourth, no celsius’.
- 🏠 Daily: tea’s fast-then-slow cooling = the exponential.
- 🏠 Daily: star colours are Wien’s law in the sky.
Quick revision
- Newton’s law of cooling: rate of cooling ∝ temperature difference with surroundings
- T(t) = T_s + (T₀ − T_s)e^(−kt) — exponential approach, never quite arriving
- Smaller ΔT → slower loss: cooling decelerates as it proceeds
- The whole series on one card: expansion, calorimetry, transfer, cooling
- Stefan’s law (radiation ∝ T⁴) and Wien’s law (peak λ ∝ 1/T) complete the picture
- Newton’s law of cooling
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