# Thermal conductivity and heat capacity

Using kinetic theory, we can derive an expression for the thermal conductivity of a gas to be
κ=nCmoleculeλ<v>/3
where n is the number density of the molecules in the gas, Cmolecule is the heat capacity of a single molcule (i.e the heat that must be given to each molecule to raise the temperature of the gas by unit temperature), λ is the mean free path and <v> is the mean speed of the molecules.

Now we can write nCmolecule=CV/V where CV is the heat capacity of the gas at constant volume and V is the total volume of the gas. Now I understand that nCmolecule=C/V where C is the heat capacity of the gas, and obviously because we have a gas we must have either C=CV or C=Cp because the gas must be held at either constant volume or constant pressure. However, I am not sure sure how to see why we have to consider the heat capacity at constant volume here - why can't it be constant pressure...

Thankyou for any answers in advance

## Answers and Replies

Bystander
Science Advisor
Homework Helper
Gold Member
why we have to consider the heat capacity at constant volume here
n is the number density
Stick with the original constraints on the calculation.

Stick with the original constraints on the calculation.

What does this mean?

Bystander
Science Advisor
Homework Helper
Gold Member
You started your calculation at constant V. Finish at constant V.

You started your calculation at constant V. Finish at constant V.

Ok, but I can't see why we started at constant V?

Bystander
Science Advisor
Homework Helper
Gold Member
You haven't included any dependence of V on P, T.

You haven't included any dependence of V on P, T.

Oh I see, so my derivation basically assumes the whole system is in a steady state (transport properties are for steady state systems), because I used a fixed temperature gradient. So then the pressure and volume must be fixed (if the pressure was fixed but volume varied, my temperature would change, so we need to fix volume and pressure).