Question

a) Kinetic theory. The speed of sound in the air is 330 m/s under standard conditions of temperature and pressure (273 K and 1 atm). Since the size of a molecule is much smaller than the average distance between the molecules, this number provides an estimate of the order of magnitude of the molecular media velocity. Consider a cubic meter of air and concentrate it on a N2 molecule that travels in the x direction at 330 m / s. Its mass is equal to 28 grams / 6.02 x 10 ^ 23.
1) Calculate the amount of movement imparted to the wall when it collides in an elastic way with it. Why is twice the amount of movement that it has?

2) How long does it take to do the 2-meter round trip between collisions against the same wall? Note that collisions between molecules (ideal gas) are not taken into account.

3) What is the average force exerted by a molecule on the wall?

4) If it is known that 6.02 x 10 ^ 23 N2 molecules (i.e. 28 grams) occupy 22.4 liters in standard conditions of temperature and pressure. How many molecules are in 1m ^ 3?

5) Assume that one third of all specific molecules average the force on the wall perpendicular to the x direction. Calculate the pressure (= force / 1m ^ 2) on the wall.

NOTE: It is expected to obtain a pressure of 1 atm (= 10 ^ 5N / m ^ 2). The calculations give a lower pressure. This is because the speed of sound underestimates the effective molecular speed by a factor of (1.4 / 3) ^ 1/2.

Answer 1

(1) mass of one N₂ molecule = ( 2810^-3 ) / ( 6.02 10²³ ) = 4.651 10^-26 kg

momentum imparted to wall = change in momentum after collision = (m v) - (-m v ) = ( 2 m v )

momentum imparted to wall = 2 4.651 10^-26 330 = 3.07 10^-23 N s

Before collision molecule has momentum mv . After collision molecule has momentum -mv

Hence to consider change of momentum , i.e., momentum imparted to wall becomes twice of initial momentum of molecule.

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Time taken for 2 m round trip = 2 / 330 = 6.1 10 ^-3 s

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average force exerted by molecule on wall = change-in-momentum/time

= 3.07 10^-23 / ( 6.1 10^-3 ) = 0.5 10^-20 N

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number of molecules in 1 m³ = 6.02 10²³ / (22.4 10^-3 ) = 0.269 10²⁶

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if 1/3 of molecules average the force, then pressure P is given by

P = (1/3) 0.269 10²⁶ 0.5 10^-20 = 0.448 10⁵ N/m²