0
$\begingroup$

I understand that the pressure exerted by a molecule on the walls of a container, in the $x$ direction is given by $P=n m v_x^2$ Where $v_x$ is the mean $x$ component of velocity, $m$ is mass and $n$ is the number of molecules.

I also understand that $\mathbf{v_x^2=v_y^2=v_z^2}$

What I'm unable to understand is this: $\mathbf{\frac{1}{3}(v_x^2+v_y^2+v_z^2)=\frac{1}{3}v^2}$
Why exactly are we multiplying the equation by $\frac{1}{3}$ ? Apparently it is to find the average. But $(v_x^2+v_y^2+v_z^2)$ gives us the resultant velocity sqaured in magnitude. Dividing it by $3$ would just be reducing its value by one-third. How do we get the average by multiplying by $\frac{1}{3}$ ?

$\endgroup$

1 Answer 1

0
$\begingroup$

It's to do with how the total velocity is worked out from the components, with Pythagoras theorem. $v^2 = {v_x}^2+{v_y}^2+{v_z}^2$

If the components are equal on average then $v^2 = 3{v_x}^2$ and ${v_x}^2 = \frac{1}{3}{v}^2$

The second formula in bold in the question is best written omitting the $\frac{1}{3}$ from each side and your first formula for P becomes $P=\frac{1}{3}mnv^2$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.