Question

Suppose that you put one air molecule on your desk, so it is in thermal equilibrium with the desk at 23.9 degree C. Suppose that there is no atmosphere to get in the way of this one molecule bouncing up and down on the desk. Calculate the typical height that the air molecule will be above your desk, so that Mgy K_B T.

Answer 1

The typical height that the air molecule will be above the desk is approximately 0.091 meters.

Step-by-step explanation:

The typical height that the air molecule will be above the desk can be calculated using the equation Mgy = k_BT, where M is the mass of the air molecule, g is the acceleration due to gravity, y is the height, k_B is the Boltzmann constant, and T is the temperature.

Since the air molecule is in thermal equilibrium with the desk, it has the same temperature as the desk, which is 23.9 degrees C or 297.05 Kelvin.

Given that the height of the desk is typically around 0.75 meters and the mass of an air molecule is approximately 4.8 x 10^-26 kilograms (using the mass of a nitrogen molecule as a proxy), substituting the known values into the equation gives:

Mgy = k_BT

(4.8 x 10^-26 kg)(9.8 m/s^2)(0.75 m) = (1.38 x 10^-23 J/K)(297.05 K)

Simplifying the equation, we can solve for y:

y = (1.38 x 10^-23 J/K)(297.05 K) / (4.8 x 10^-26 kg)(9.8 m/s^2)

y ≈ 0.091 meters

Therefore, the typical height that the air molecule will be above the desk is approximately 0.091 meters.