Origin of Pressure

Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container.

Learning Objective

Express the relationship between the pressure and the average kinetic energy of gas molecules in the form of equation

Key Points

We can gain a better understanding of pressure (and temperature as well) from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.
Pressure, a macroscopic property, can be related to the average (translational) kinetic energy per molecule which is a microscopic property by $P = \frac{nm\overline{v^2}}{3}$ .
Since the assumption is that the particles move in random directions, the average value of velocity squared along each direction must be same. This gives: $\overline{v_x^2} = \overline{v_y^2} =\overline{v_z^2} =\overline{v^2}/3$ .

Terms

Newtonian mechanics
Early classical mechanics as propounded by Isaac Newton, especially that based on his laws of motion and theory of gravity.
kinetic theory of gases
The kinetic theory of gases describes a gas as a large number of small particles (atoms or molecules), all of which are in constant, random motion.

Full Text

In Newtonian mechanics, if pressure is the force divided by the area on which the force is exerted, then what is the origin of pressure in a gas? What forces create the pressure? We can gain a better understanding of pressure (and temperature as well) from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.

Microscopic Origin of Pressure

Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container, as illustrated in the figure below. Consider a gas of N molecules, each of mass m, enclosed in a cubical container of volume V=L³. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall ($\Delta p$) is:

Translational Motion of Helium

Real gases do not always behave according to the ideal model under certain conditions, such as high pressure. Here, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure.

$\begin{aligned} \Delta p &= p_{i,x}-p_{f,x} = p_{i,x}-(-p_{i,x})\\ &= 2p_{i,x} = 2mv_x \end{aligned}$

where v_x is the x-component of the initial velocity of the particle.

The particle impacts one specific side wall once every $\Delta t = \frac{2L}{v_x}$,

(where L is the distance between opposite walls). The force due to this particle is:

$\displaystyle F = \frac{\Delta p}{\Delta t} = \frac{mv_x^2}{L}$.

The total force on the wall, therefore, is:

$\displaystyle F = \frac{Nm\overline{v_x^2}}{L}$,

where the bar denotes an average over the N particles. Since the assumption is that the particles move in random directions, if we divide the velocity vectors of all particles in three mutually perpendicular directions, the average value of the squared velocity along each direction must be same. (This does not mean that each particle always travel in 45 degrees to the coordinate axes. )

This gives $\overline{v_x^2} = \overline{v^2}/3$. We can rewrite the force as $F=\frac{Nm\overline{v^2}}{3L}$.

This force is exerted on an area L². Therefore the pressure of the gas is:

$\displaystyle P = \frac{F}{L^2} = \frac{Nm\overline{v^2}}{3V} = \frac{nm\overline{v^2}}{3}$,

where V=L³ is the volume of the box. The fraction n=N/V is the number density of the gas. This is a first non-trivial result of the kinetic theory because it relates pressure (a macroscopic property) to the average (translational) kinetic energy per molecule which is a microscopic property.

Pressure

Pressure arises from the force exerted by molecules or atoms impacting on the walls of a container.

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