Newton’s Law of Cooling 🌡️

1. Cool-down story

Imagine pouring hot water into a cup. At first the water steams, but minute by minute it cools until it matches the room temperature. The bigger the temperature gap, the faster the drop! 🔥➡️❄️ :contentReference[oaicite:0]{index=0}

2. The Law in One Line

Newton’s law of cooling says the rate at which an object loses heat is directly proportional to the temperature difference between the object (\(T_2\)) and its surroundings (\(T_1\)) when that difference is small:

\[ -\frac{dQ}{dt}=k\,(T_2-T_1) \] :contentReference[oaicite:1]{index=1}

Here \(k\) depends on the object’s surface area and how easily the surface lets heat through. :contentReference[oaicite:2]{index=2}

3. Turning heat into temperature drop

For a mass \(m\) with specific heat capacity \(s\) you write the lost heat as \(dQ = m s\,dT_2\). Put this into the law and you get a sweet differential equation: ⏱️

\[ m s\,\frac{dT_2}{dt} = -k\,(T_2-T_1) \] :contentReference[oaicite:3]{index=3}

Divide both sides by \(m s\) and call \(\displaystyle K=\frac{k}{m s}\):

\[ \frac{dT_2}{dt} = -K\,(T_2-T_1) \] :contentReference[oaicite:4]{index=4}

4. Exponential cooling curve 📉

Integrate and you discover an exponential drop:

\[ \ln\!\bigl(T_2-T_1\bigr) = -K t + c \quad\Longrightarrow\quad T_2 = T_1 + C’\,e^{-K t} \]

The constant \(C’\) depends on the starting temperature. Draw \(\ln(T_2-T_1)\) versus time and you should see a straight line falling to the right – a neat experimental check! 📈 :contentReference[oaicite:5]{index=5}

5. Quick lab activity 🧪

• Heat 300 mL of water about \(40^{\circ}\text{C}\) above room temperature in a calorimeter.
• Start a stopwatch and read the temperature every minute while gently stirring.
• Plot \(\Delta T = T_2 – T_1\) against time to watch the curve bend downwards.
• You will notice the steepest fall at the beginning; the curve flattens as the water cools. 👍

This simple setup shows how the rate slows as the temperature gap narrows. :contentReference[oaicite:6]{index=6}

6. Everyday example

A pan of hot food drops from \(94^{\circ}\text{C}\) to \(86^{\circ}\text{C}\) in 2 minutes when the room sits at \(20^{\circ}\text{C}\). Using the same \(K\), it will slip from \(71^{\circ}\text{C}\) to \(69^{\circ}\text{C}\) in just 42 seconds. 🕒 :contentReference[oaicite:7]{index=7}

7. Radiative cooling bonus 🌞

Objects also cool by radiation. A perfect radiator with area \(A\) and temperature \(T\) throws out energy every second according to the Stefan–Boltzmann expression:

\[ H = A\,\sigma\,T^{4} \] :contentReference[oaicite:8]{index=8}

If the surface isn’t perfect, multiply by its emissivity \(e\) (for shiny tungsten \(e\approx 0.4\), for skin \(e\approx 0.97\)):

\[ H = e\,A\,\sigma\,T^{4} \]

When the surroundings rest at \(T_s\), the net radiative loss becomes

\[ H = e\,\sigma\,A\,(T^{4} – T_s^{4}) \] :contentReference[oaicite:9]{index=9}

The Stefan–Boltzmann constant is \(\sigma = 5.67 \times 10^{-8}\,\text{W\,m}^{-2}\text{K}^{-4}\). 💡 :contentReference[oaicite:10]{index=10}

8. Why should you care? 🎓

Engineers design radiators, refrigerators, and even shiny space blankets by playing with these cooling laws. Understanding the exponential drop helps you estimate how long your tea stays hot during revision sessions! ☕

High-Yield Ideas for NEET 🎯

1. Newton’s law of cooling equation \(-\dfrac{dQ}{dt}=k\,(T_2-T_1)\) and its validity for small temperature differences. :contentReference[oaicite:11]{index=11}
2. Exponential temperature–time relation \(T_2 = T_1 + C’e^{-K t}\) and the straight-line check with \(\ln(T_2-T_1)\) vs \(t\). :contentReference[oaicite:12]{index=12}
3. Stefan–Boltzmann law \(H=A\sigma T^{4}\) and net form \(H=e\sigma A\,(T^{4}-T_s^{4})\). :contentReference[oaicite:13]{index=13}
4. Emissivity (\(e\)) and its effect on cooling (lamp black ≈1, tungsten ≈0.4, skin ≈0.97). :contentReference[oaicite:14]{index=14}
5. Example-based numerical problems using the same \(K\) to predict new cooling times. :contentReference[oaicite:15]{index=15}

Keep practicing these ideas, and you’ll breeze through the heat-transfer questions! 🚀