Question

Use the formula v = √γP/ρ to explain why the speed of sound in air
(a) is independent of pressure,
(b) increases with temperature,
(c) increases with humidity.

Answer 1

a) we know ,
p =d RT/M
where d is density , T is temperature , R is universal constant , M is molar mass and p is the pressure .
we see p is directly proportional to d
it means if we increase p value then is also increase . but in sound both term included so, if we increase p then no change in velocity of sound .

b) v=root {YP/d}
put P=dRT/M
then ,
velocity of sound =root {dRT/M}
here we velocity of sound is directly proportional to square root of temperature .

c) humidity :- if humidity increase , then density decrease ,hence speed of sound increase .we can say that if humidity increase velocity of sound also increases.

Answer 2

Answer:

(a) Take the relation:

v = √γP/ρ ….(i)

where,

Density, ρ = Mass/Volume = M/V

M = Molecular weight of the gas

V = Volume of the gas

Hence, equation (i) reduces to:

v = √γPV/M ….(ii)

Now from the ideal gas equation for n = 1:

PV = RT

For constant T, PV = Constant

Since both M and γ are constants, v = Constant

Hence, at a constant temperature, the speed of sound in a gaseous medium is independent of the change in the pressure of the gas.

(b) Take the relation:

v = √γP/ρ ….(i)

For one mole of any ideal gas, the equation can be written as:

PV = RT

P = RT/V ….(ii)

Substituting equation (ii) in equation (i), we get:

v = √γRT/Vρ = √γRT/M …..(iii)

where,

mass, M = ρV is a constant

γ and R are also constants

We conclude from equation (iii) that v ∝ √T

Hence, the speed of sound in a gas is directly proportional to the square root of the temperature of the gaseous medium, i.e., the speed of the sound increases with an increase in the temperature of the gaseous medium and vice versa.

(c) Let vm and vd be the speed of sound in moist air and dry air respectively.

Let ρm and ρdbe the densities of the moist air and dry air respectively.

However, the presence of water vapour reduces the density of air, i.e.,

ρd < ρm

∴ vm > vd

Hence, the speed of sound in mois air is greater than it is in dry air. Thus, in gaseous medium, the speed of sound increases with humidity.