Question

What is the formula of pressure according to kinetic theory in terms of mass density ?

In my book, the formula given is `P= 1/3 nm(v^2)_(av)` where n is number density, m is mass of one molecule and

`(v^2)_(av)` is average of square of all speeds in random direction. Now, if I were to write formula for P in terms of mass density i.e `rho = m/V`, it should be

`P= 1/3Nrho(v^2)_(av)` where N is number of molecules but the formula given in my book for this case is

`P=1/3rho(v^2)_(av)`. What is the correct formula?

Answer 1

There is error in calculation of `rho`.
In the equation (1) `m` is defined as mass of one molecule. While calculating `rho`, `m` has been taken as total mass of gas sample.

Explanation:

Given

`P= 1/3 nmbar(v^2)` .....(1)
where `n` is number density, `m` is mass of one molecule and `bar(v^2)` is average of square of all speeds in random direction.

Now, writing formula for `P` in terms of mass density `rho` we see that

`rho = "Total mass"/"Volume"`

`=>rho = nm`
`n` being number density already stated.

Inserting in (1) we get

`P= 1/3rhobar(v^2)`