Animated Solution for Physics - Properties of Solids and Liquids: Dimensional formula for thermal conductivity is (Here, $K$ denotes the temperature)

Imagine you are holding a metal rod over a campfire. Very quickly, the end you are holding becomes unbearably hot. This everyday phenomenon is a perfect demonstration of heat conduction. But how do physicists quantify a material's ability to conduct heat? They use a property called thermal conductivity, denoted by the symbol $K$.

In this journey, we aren't just going to memorize the dimensional formula for thermal conductivity; we are going to build it from the ground up using the fundamental laws of physics.

The Master Equation

To find the dimensions of any physical quantity, we first need an equation that connects it to things we already know. For thermal conductivity, our golden ticket is Fourier's Law of Heat Conduction.

Fourier's Law states that the rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area, at right angles to that gradient, through which the heat flows. Mathematically, it is expressed as:

Here, $\frac{dQ}{dt}$ is the rate of heat flow, $A$ is the cross-sectional area, $dT$ is the temperature difference, and $dx$ is the thickness or length of the material. Our goal is to isolate our hero, $K$.

Let's rearrange the equation to make $K$ the subject. By cross-multiplying, we get:

Now, we have $K$ expressed entirely in terms of measurable physical quantities. The next step is to substitute the SI units for each of these quantities to see what we are dealing with.

Let's plug in the standard SI units: - Heat ($dQ$) is a form of energy, measured in Joules (J). - Distance ($dx$) is measured in meters (m). - Area ($A$) is measured in square meters (m${}^2$). - Time ($dt$) is measured in seconds (s). - Temperature ($dT$) is measured in Kelvin (K).

Substituting these into our rearranged equation gives:

We can immediately simplify this by canceling one meter from the numerator and denominator:

We are almost there, but we have a slight problem: the Joule is a derived unit, not a fundamental one. To find the true dimensional formula, we must break the Joule down into its fundamental components (mass, length, and time).

Recall that Work = Force $\times$ Displacement. Therefore, $1\text{ Joule}=1\text{ kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-2}$. Let's substitute this expanded definition of the Joule back into our unit equation:

Now, we perform the final cancellation. One meter cancels out, and the seconds combine in the denominator:

Finally, we translate these fundamental units into their corresponding dimensional symbols: Mass ($M$), Length ($L$), Time ($T$), and Temperature ($K$). Bringing all terms to the numerator, we arrive at our grand conclusion:

And there you have it! By simply following the trail of units left by Fourier's Law, we have successfully derived the dimensional formula for thermal conductivity.