Animated Solution for Physics - Thermodynamics: If $10^{22}$ gas molecules each of mass $10^{-26}$ kg collide with a surface (perpendicular to it) elastically per second over an area $1{\text{m}}^2$ with a speed $10^4$ m/s, the pressure exerted by the gas molecules will be of the order of

The kinetic theory of gases provides a beautiful bridge between the microscopic world of atoms and the macroscopic properties we can measure, like pressure and temperature. In this problem, we are going to calculate the pressure exerted by a stream of gas molecules bombarding a surface.

Imagine you are standing next to a wall, and a continuous stream of tiny invisible particles—gas molecules—is colliding with it. Each molecule has a mass $m$ and is moving with a velocity $v$ perpendicular to the wall.

When a molecule hits the wall perfectly elastically, it doesn't lose any kinetic energy. It simply reverses its direction. It approaches with momentum $mv$ and rebounds with momentum $-mv$.

The change in momentum for a single molecule is:

According to Newton's Second Law of Motion, force is the rate of change of momentum. If $n$ molecules collide with the wall every second, the total change in momentum per second gives us the total force $F$ exerted on the wall:

Pressure $P$ is defined as the force applied per unit area $A$. Therefore, the pressure exerted by the gas molecules is:

Now, let's bring in the numbers given in the problem. We have: - Mass of each molecule, $m=10^{-26}\text{ kg}$ - Number of collisions per second, $n=10^{22}{\text{ s}}^{-1}$ - Speed of molecules, $v=10^4\text{ m/s}$ - Area of the surface, $A=1{\text{ m}}^2$

Substituting these values into our pressure equation:

Let's carefully combine the powers of 10:

So, the powers of 10 perfectly cancel out!

The question asks for the order of magnitude of the pressure. Since $2$ is a number between $1$ and $10$ (specifically, less than $\sqrt{10}$), the order of magnitude is $10^0$.

Interestingly, none of the options provided in the original exam ($10^4$, $10^8$, $10^3$, $10^{16}$) match our correct answer. This makes it a bonus question where all given options are incorrect!