# An Expression for $\Delta T$ for a Thermal Conductivity derivation

In kinetic theory, the thermal conductivity of a gas is $$K = \frac{1}{3} C vl.$$ In deriving this formula, why is the following equation valid?

"Now $$\Delta T$$ between the ends of a free path of the particle is given by $$\Delta T = \frac{dT}{dx}l_x$$ where $$\tau$$ is the average time between collisions."

This is found in Kittel, chapter 5. Thank you!

If you assume that there is a uniform temperature gradient $$\text{d}T/\text{d}x$$ along some path of length $$l_x$$ then the temperature difference between the two ends of the path is $$\text{d}T/\text{d}x \cdot l_x$$.