How Does the Rate of Thermal Conduction Depend on Material and Temperature Gradient?

The rate of thermal conduction increases with greater thermal conductivity and a larger magnitude of the temperature gradient. Thermal energy is transferred from regions of higher temperature to regions of lower temperature.

The Conduction Equation

In IB Physics B.1 Thermal energy transfers, this relationship applies at both SL and HL:

$$\frac{\Delta Q}{\Delta t}=-kA\frac{\Delta T}{\Delta x}$$

Here, $\frac{\Delta Q}{\Delta t}$ is the rate of thermal energy transfer in watts, $k$ is thermal conductivity in $\text{W m}^{-1}\text{K}^{-1}$, is cross-sectional area in , and is the temperature gradient in .

The negative sign indicates that energy flows down the temperature gradient. If only the magnitude is required, use

$$\left|\frac{\Delta Q}{\Delta t}\right|=kA\frac{|\Delta T|}{\Delta x}.$$

At particle level, conduction occurs through interactions between neighboring particles; in metals, mobile electrons also transfer energy rapidly. Therefore, metals generally have larger conductivity than insulating materials, while the gradient controls the driving difference across each unit of distance in the material.

Worked Example

A slab has $k=0.80\,\text{W m}^{-1}\text{K}^{-1}$, $A=0.020\,\text{m}^2$, thickness , and temperature difference :

$$\left|\frac{\Delta Q}{\Delta t}\right|=(0.80)(0.020)\frac{30}{0.050}=9.6\,\text{W}.$$

If the thickness were halved while all other quantities remained constant, the gradient and conduction rate would double to $19.2\,\text{W}$.

A common misconception is that conduction rate depends only on temperature difference. It depends on the temperature difference per unit distance, so the same difference across a thinner slab gives a greater rate. The equation assumes steady one-dimensional conduction through a uniform material with constant $k$.

Exam Technique

For a Calculate question, show the equation, substitution, and final value. For an Explain question, connect greater $k$, larger $A$, or a steeper gradient to increased transfer rate. Do not interpret the negative sign as a negative amount of energy; it specifies the direction of transfer.