Study Exponential Temperature Decay and Heat Transfer
✅ FREE Experiment • 📊 Real-Time Graphs • 🎓 NEB Class 11 Physics
Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in temperature between the body and its surroundings, provided the temperature difference is small. Mathematically, this leads to an exponential decay of temperature with time.
This experiment simulates a hot object (like a cup of coffee or hot water) cooling down to room temperature. You'll observe the temperature decrease over time, plot cooling curves, calculate the cooling constant (k), and verify the exponential nature of the cooling process. This principle is fundamental to understanding heat transfer in everyday situations.
Watch the mercury level drop as temperature decreases exponentially
Temperature of hot object at start
Room/surrounding temperature
Higher k = faster cooling (depends on material & surface area)
Interval for recording observations
Where: T(t) = temperature at time t, T₀ = ambient temperature, T_i = initial temperature, k = cooling constant, t = time
| S.No. | Time (s) | Temperature (°C) | T - T₀ (°C) | ln(T - T₀) |
|---|---|---|---|---|
| No observations recorded yet. Click "Record" to add readings | ||||
Linear Verification: Plot ln(T - T₀) vs time. If Newton's Law holds, you'll get a straight line with slope = -k. The y-intercept gives ln(T_i - T₀).
Exponential decay curve. Temperature asymptotically approaches ambient temperature.
Linear plot for verification. Slope = -k (cooling constant).
Rate of cooling (dT/dt) decreases as temperature approaches ambient.
Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature:
Where:
Solving the differential equation dT/dt = -k(T - T₀):
The cooling constant k determines how fast an object cools. It depends on:
If we know T at two different times t₁ and t₂:
Taking natural log of both sides:
ln(T - T₀) = ln(T_i - T₀) - kt
This is a linear equation (y = mx + c form) where:
Plot ln(T - T₀) vs t. The slope of the best-fit line gives -k.
The "half-life" is the time for temperature difference to reduce by half:
Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in temperature between the body and its surroundings, provided the temperature difference is small. Mathematically: dT/dt = -k(T - T₀).
T(t) = T₀ + (T_i - T₀)e^(-kt), where T(t) is temperature at time t, T₀ is ambient temperature, T_i is initial temperature, and k is the cooling constant. This shows exponential decay toward ambient temperature.
The cooling constant k characterizes how fast an object cools. Higher k means faster cooling. It depends on surface area, thermal conductivity, specific heat capacity, mass, and the surrounding medium. Units are typically s⁻¹ or min⁻¹.
Plot ln(T - T₀) versus time. If Newton's Law holds, this plot will be a straight line. The slope of this line equals -k (negative of cooling constant). This linear relationship confirms the exponential decay predicted by the law.
As T approaches T₀, the temperature difference (T - T₀) becomes smaller, so the cooling rate dT/dt = -k(T - T₀) also becomes smaller. The object continues cooling but at an ever-decreasing rate, theoretically never exactly reaching T₀ (but practically does).
k increases with: (1) Larger surface area, (2) Better thermal conductivity, (3) Lower specific heat capacity, (4) Smaller mass, (5) Air movement (convection), (6) Lower surrounding pressure. k = hA/(mc), where h is heat transfer coefficient, A is area, m is mass, c is specific heat.
(1) Temperature difference is relatively small (typically < 100°C), (2) Ambient temperature is constant, (3) Object has uniform temperature throughout, (4) Heat loss is primarily by convection, not radiation, (5) No phase changes occur, (6) Object properties don't change with temperature.
Forensic scientists estimate time of death by measuring body temperature and using Newton's Law. Normal body temperature is 37°C. By measuring current body temperature and knowing ambient temperature, they can calculate time since death using T(t) = T₀ + (37 - T₀)e^(-kt), with known k for human bodies.
For large temperature differences, radiation becomes significant (Stefan-Boltzmann Law: radiation ∝ T⁴). Newton's Law assumes convection dominates, which is valid when radiation is negligible. This occurs when temperature differences are relatively small (< 100°C approximately).
The half-life t_½ is the time required for the temperature difference (T - T₀) to reduce to half its initial value. It equals t_½ = ln(2)/k ≈ 0.693/k. Like radioactive decay, this half-life is constant regardless of initial temperature difference.
Stirring increases convection within the liquid, making temperature more uniform and bringing hotter liquid to the surface faster. This effectively increases k (cooling constant), making cooling faster. Without stirring, surface cools faster than interior, violating uniform temperature assumption.
Yes, the same law applies to heating. If an object is cooler than surroundings, it warms up following: T(t) = T₀ + (T_i - T₀)e^(-kt). Here, if T_i < T₀, the object temperature increases exponentially toward T₀. The same constant k applies for both heating and cooling.
dT/dt is the rate of temperature change. The negative sign means temperature decreases (for cooling). The proportionality to (T - T₀) means cooling rate is faster when the temperature difference is larger. When T = T₀, dT/dt = 0 (no more cooling).
Larger surface area increases heat loss rate, increasing k. A flat, spread-out object cools faster than a compact sphere of the same volume. This is why elephants have large ears (increase surface area for cooling) and why radiators have fins.
If T₀ changes, Newton's Law as stated doesn't apply accurately. The derivation assumes constant T₀. In practice, ensure room temperature stays constant by avoiding drafts, sunlight, and other heat sources. Small fluctuations (±1°C) are usually acceptable.