Newton's Law of Cooling

Newton’S Law Of Cooling

When a hot object is placed in a cooler environment, it loses heat to the surroundings and cools down. Newton's Law of Cooling describes the rate at which this cooling occurs under specific conditions.

---

Statement of Newton's Law of Cooling

Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the temperature difference between the body and its surroundings, provided the temperature difference is small and the nature of the exposed surface remains the same.

Mathematically, the rate of cooling ($ \frac{dT}{dt} $, the rate of change of temperature) or the rate of heat loss ($ \frac{dQ}{dt} $) is proportional to the temperature difference:

$ \frac{dQ}{dt} \propto (T - T_s) $

where $T$ is the temperature of the body and $T_s$ is the temperature of the surroundings (assuming $T > T_s$). The rate of heat loss is negative if $T < T_s$ (body gains heat and warms up). It is often stated as the rate of change of temperature:

$ \frac{dT}{dt} \propto -(T - T_s) $

Introducing a constant of proportionality $K$ (a positive constant that depends on the surface area, nature of the surface, and properties of the surroundings):

$ \frac{dT}{dt} = -K (T - T_s) $

The rate of heat loss is also related to the rate of change of temperature by $dQ = mc \, dT$, where $m$ is the mass and $c$ is the specific heat capacity of the body. So, $\frac{dQ}{dt} = mc \frac{dT}{dt}$.

$ mc \frac{dT}{dt} = -A' (T - T_s) $

Here, $A'$ is another constant that incorporates the surface properties and surrounding conditions. This is often written as:

$ \frac{dT}{dt} = -\frac{A'}{mc} (T - T_s) = -k (T - T_s) $

where $k = \frac{A'}{mc}$ is the cooling constant, which depends on the properties of the body (mass, specific heat capacity) and the properties of the surroundings and surface area.

---

Conditions for Validity

Newton's Law of Cooling is an approximation and is valid under certain conditions:

• The temperature difference between the body and the surroundings is small ($T - T_s$ is small).
• Heat is lost mainly by convection and radiation, and convection currents are well-established.
• The surface area and nature of the surface exposed to the surroundings remain constant during cooling.
• The temperature of the surroundings ($T_s$) remains constant.

For larger temperature differences, the rate of heat loss by radiation is proportional to $(T^4 - T_s^4)$, which is not linear in $(T - T_s)$, so Newton's law becomes less accurate. However, for small temperature differences, $T^4 - T_s^4 \approx 4 T_s^3 (T - T_s)$, which is linear in $(T - T_s)$, thus validating the linear relationship in that limit.

---

Solving the Differential Equation

The equation $\frac{dT}{dt} = -k (T - T_s)$ is a first-order linear differential equation. Its solution gives the temperature of the body as a function of time:

$ T(t) = T_s + (T_0 - T_s) e^{-kt} $

where $T_0$ is the initial temperature of the body at $t=0$, and $T_s$ is the constant temperature of the surroundings. This equation shows that the temperature difference $(T - T_s)$ decreases exponentially with time, and the body asymptotically approaches the temperature of the surroundings.

(Image Placeholder: A graph with Time on the x-axis and Temperature on the y-axis. Show a decreasing exponential curve starting from an initial temperature T0 above the surrounding temperature Ts. The curve approaches the horizontal line at Ts asymptotically.)

---

Experimental Determination of Cooling Constant (k)

The cooling constant $k$ (or the rate of cooling) can be determined experimentally by recording the temperature of a hot object placed in a constant-temperature environment at regular time intervals. A graph of temperature versus time can be plotted, and the slope $\frac{dT}{dt}$ at various temperatures can be found. Alternatively, a graph of $\frac{dT}{dt}$ versus $(T - T_s)$ should yield a straight line passing through the origin with a slope of $-k$ (or $k$ if plotting $|dT/dt|$ vs $T-Ts$).

The rate of cooling over a finite temperature interval from $T_1$ to $T_2$ in time $\Delta t$ can be approximated as the average rate of change, $\frac{T_1 - T_2}{\Delta t}$, being proportional to the average temperature difference, $\frac{T_1 + T_2}{2} - T_s$.

$ \frac{T_1 - T_2}{\Delta t} = K' \left(\frac{T_1 + T_2}{2} - T_s\right) $

This approximate form is often used in laboratory experiments to verify Newton's Law of Cooling or determine the cooling constant $K'$.

---

Applications

• Predicting Cooling Times: Used to estimate how long it takes for objects to cool down, important in manufacturing processes, food science, etc.
• Forensics: Estimating the time of death by measuring the body temperature and using cooling rate models (though biological processes and varying conditions make this complex).
• Designing Cooling Systems: Provides a basis for understanding and designing systems where cooling is involved.

Answer: