Speed of Sound in Gas According to Laplace’s correction

 The speed of sound in air is given by the relation below:

                                   V = √є/d  

Where є is the bulk modulus of the gas, d is the density

Newton was the first to carry out this calculation but the result was far below the experimental value of 330m/s. His error was later corrected by Laplace in1816, Laplace suggested that є should be the adiabatic bulk modulus of gas not, its isothermal bulk as assumed by Newton. The adiabatic bulk modulus of gas is β = µp where p is pressure and µ is the ratio of molar heat capacities. The speed of sound becomes 

                                                V = √µp/d

From the ideal gas equation,

                                                  PV = RT

Where R is the molar gas constant, T is the absolute temperature and v is the molar volume of a gas. Then,

                                                   P = RT/V

Note: density = M/V

Where M is molecular mass. So the velocity of sound in gas then becomes

                                                    V = √µRT/M

Since µ, R and M are constants for any given gas, it shows that the velocity of sound in a gas is independent of pressure at a constant temperature.

 Experiments have also shown that the velocity of sound is proportional to the square root of its absolute temperature 

                                                       V ∞ √T

 Where T is the absolute temperature.

  Example Question 1

A small piece of cork in a ripple tank oscillates up and down as ripples pass it. If the ripples travel at 4m/s and have a wavelength of 15m and an amplitude of 10m, what is the maximum velocity of the cork?

Solution

Velocity of the ripples = 4m/s

Wavelength = 15m

Amplitude = 10m

Velocity of the cork = ??

The maximum velocity of the cork is given by

                                Vm =      

2πa/T

Where a is the amplitude and T is the period.

 Since the period is not given, to find the T

          T = 1/f

We know that the frequency is the same for two oscillating bodies.

  V = fλ

.λ = 15m

V = 4m/s

.f = ?

   .f = 4÷ 15 = 0.267Hz

 T = 1/0.267 = 3.75sec.

Maximum velocity of the cork

V = 2πa ÷ T = 2x 3.143 x 10÷ 3.75 = 16.8m/s  

Example question 2

A source of sound of frequency 660Hz emits waves of wavelength 1200mm in air at 400C. What is the velocity of sound in air at this temperature? What would be the wavelength of the sound from the source in air at 00C?

Solution 

1. .f = 660Hz

            .λ = 1200mm = 1.2m

            Velocity of sound at 400C = .fλ = 660 x 1.2 = 792m/s

1. Velocity of sound ∞ √T in kelvin

                                V/V0 = √T/T0  

V at 40C = 792m/s

V0 at 00C = ??

T = 273+40 = 313K

T0 = 273 + 0 = 273K

792/V0 = √313/273

792/V0 = √1.1465

792/V0 = 1.07076/1

V0 = 792/1.07076 = 739.7m/s   

or 740m/s

Wavelength at 00C

V = fλ  

740m/s = 660 x λ

.λ = 740/660  

= 1.12m

Example question 3

The speed of sound in air at 00C is 330m/s. calculate the speed at a temperature of 450C

Solution

V at 450C =??

V0 at 00C = 330m/s

T = 273 +45 = 318K

T0 = 273 + 0 = 273K

V0/V = √273/318

330/V = √273/318

330/V = √0.8585

330/V = 0.9266

V = 330/0.9266 = 

356.2m/s or 356m/s