Name: Introduction to Brownian Motion
Uploaded: 2019-07-19
Duration: 3 min 35 s
Description: The study of Brownian motion is important in the study of the kinetic theory of gases. But what is meant by the Brownian motion? Let's understand it in this...

Question

Write down an expression for the pressure of an ideal gas.

Accepted Answer

Let A₁and A₂ be the parallel faces perpendicular to the x-axis. Consider a molecule moving with velocity v in the container. The components of velocity along the axes are v_x, v_y, and v_z. Now, when the molecule is colliding with face A₁, the x component of velocity is reversed while the y and z component of velocity remains unchanged.

The change in momentum of the molecule is:

$Δ P = (- m v x) - (m v x) = - 2 m v x \dots \dots \dots . . (1)$

Since the momentum remains conserved, the change in momentum of the wall is 2mvx

After the collision, the molecule travels towards the face A2 with x component of the velocity equal to −vx

travelled from $A_{1} t o A_{2} = L$

Therefore, time = $\frac{L}{x}$

After collision with A2, the molecule again travels to A1. Thus, the time between two collisions is 2Lvx

Therefore, the number of collisions of the molecule per unit time:

$n = \frac{V_{x}}{2 L} . . . . . . . . (2)$

Using equations 1 and 2,

The momentum imparted per unit time to the wall by the molecule:

$Δ F = n Δ p = \frac{m}{L} {v_{x}}^{2}$

Therefore, the total force on wall A1 due to all the molecules is

$F = Ʃ \frac{m}{L} v x^{2} = \frac{m}{L} Ʃ v x^{2} N o w, Ʃ v x^{2} = Ʃ v y^{2} = Ʃ v z^{2} = \frac{1}{3} Ʃ v^{2} T h e r e f o r e, F = \frac{1}{3} m L Ʃ v^{2}$

Pressure is force per unit area so that

$P = \frac{F}{L^{2}}$

$= \frac{1}{3} \frac{M}{L^{3}} \sum v^{2}$

$= \frac{1}{3} ρ \sum v^{2} N$

Where,

M= total mass of gas

ρ = density of the gas

Now, ∑v2N is written as v2 and is called the mean square speed.
$P = \frac{1}{3} ρ ν^{2}$

The expression for the pressure of an ideal gas is

$P = \frac{1}{3} ρ v^{2}$

$ρ$ -density the ideal gas in the container

$V^{2}$ -is the mean square velocity of the molecules

Write down an expression for the pressure of an ideal gas.

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