Answered: How does the concept of pressure follow…

Related questions

Question

How does the concept of pressure follow from the kinetic molecular theory?

Expert Solution

Step 1

This is a problem from the kinetic theory of gases.

According to the kinetic theory, the pressure due to the ideal gas in a cubical container can be written as,

$P = \frac{1}{3} \frac{N}{V} m v^{2} W h e r e, N - N u m b e r o f m o l e c u l e s m - M a s s o f a e a c h m o l e c u l e v^{2} - M e a n o f t h e s q u a r e o f v e l o c i t i e s o f a l l t h e m o l e c u l e s$

We will prove this in the next step.

Step 2

Let us consider a cubic box having the sides of length L containing an ideal gas inside it.

Consider a typical molecule, moving in positive x-direction with a speed v and which going to collide with the wall parallel to the yz-plane and bounce back (Consider this to be an elastic collision).

The components of velocities are v_x , v_y and v_z .

The change in momnetum of a molecule is given by,

$∆ p_{x} = - m v_{x} - m v_{x} = - 2 m v_{x} . . . [m v b e f o r e c o l l i d i n g a n d - m v a f t e r c o l l i d i n g]$ ...(1)

Time taken by the molecule between the two successive collisions can be written as,

$∆ t = \frac{2 L}{v_{x}} . . . [d i s \tan c e i n 2 L a n d v e l o c i t y i s v_{x}]$ ...(2)

Now, The force exerted by a single molecule on the wall of a container is given by the,

$F = \frac{∆ p}{∆ t}$ ....(3)

Substituting the equation 1 and 2 in equation 3, we get,

$F = \frac{2 m v_{x_{1}}}{\frac{2 L}{v_{x_{1}}}} = \frac{m {v_{x_{1}}}^{2}}{L}$ ...(4)

Step by step

Solved in 3 steps