Question

(a) Compute the derivative of the speed of sound in air with respect to the absolute temperature, and show that the differentials dv and dT obey dv/v = 1/2 dT/T. (b) Use this result to estimate the percentage change in the speed of sound when the temperature changes from 0�C to 24.8�C. (c) If the speed of sound is 332 m/s at 0�C, estimate its value at 24.8�C using the differential approximation. (d) How does this approximation compare with the result of an exact calculation? (Enter the value from the exact calculation.)

Answer 1

The speed of sound in a gas is given by v = sqrt(γRT/M)

where,
R is the gas constant,
T is the absolute temperature,
M is the molecular mass of the gas,
γ is a constant

(a)
dv/dT = d/dT[sqrt(γRT/M)]
dv/dT = 1/2 * sqrt(M/γRT) * (γR/M)
dv/dT = 1/2 * v/T

Rearranging -

dv/v = 1/2 dT/T

Hence Proved.

(b)
dT = 24.8 - 0 = 24.8

Substituing Values in above Expression -

dv/v = 1/2 * (24.8/273)
dv/v = 0.0454

Percentage change = 4.5%

(c)
Speed of sound at 0^o C = 332 m/s

V_24.8 = v₀ * (1+dv/v)
V_24.8 = 332 * (1 + 0.0454)
V_24.8 = 347.1 m/s

(d)
v = sqrt(γRT/M)
V_24.8 /v₀ = sqrt(γRT_24.8/M) / sqrt(γRT₀/M)
V_24.8/332 = sqrt(297.95/273.)
V_24.8 = 332 * sqrt(297.95/273.) m/s
V_24.8 = 346.84 m/s

We Can clearly see the two values are nearly approximate.