Problems and Solutions on Thermodynamics and Statistical Mechanics

366 Problems^8 Sdutiom on Thermodynamics # statistical Mechanics

2176

The speed of sound (cg) in a dilute gas like air is given by the adiabatic

compressibility:

c; = [ (3,I-l = rz kT

where M is the mean molecular weight, k is Boltzmann's constant and 7 is

the ratio of principal specific heats.

(a) Estimate numerically for air at room temperature,

(1) the speed of sound;

(2) the mean molecular collisions frequency;

(3) the molecular mean free path;

(4) the ratio of mean free path to typical wave length;

(5) the ratio of typical wave frequency to collision frequency. (Use

(b) Use the ratios found above to explain why adiabatic conditions are

(UC, Berkeley)

Y = 300 Hz as typical wave frequency.)

relevant for sound.

Solution:

(a) (1) c, = =^350 m/s.

(2), (3) The mean free path is

The mean collision frequency is

• V
  f = - = 1.2 x lo8 s-I.
  1

1 IY

A c,

(4) - = - = 3.4 x 10-6.

(5) v/f = 2.5 x lo-'.
(b) As sound waves compress the air in a scale of the wavelength A,
we shall estimate the ratio of the time for heat, which is transferred by
the motion of the molecules, to travel the distance X to the period of the