(a) Compute the derivative of the speed of sound in air with respect to the abso

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 0C to 24.8C. (c) If the speed of sound is 332 m/s at 0C, estimate its value at 24.8C using the differential approximation. (d) How does this approximation compare with the result of an exact calculation? (Enter the value from the exact calculation.)

Explanation / Answer

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

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

(a)
dv/dT = d/dT[sqrt(RT/M)]
dv/dT = 1/2 * sqrt(M/RT) * (R/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 0o C = 332 m/s

V24.8 = v0 * (1+dv/v)
V24.8 = 332 * (1 + 0.0454)
V24.8 = 347.1 m/s

(d)
v = sqrt(RT/M)
V24.8 /v0 = sqrt(RT24.8/M) / sqrt(RT0/M)
V24.8/332 = sqrt(297.95/273.)
V24.8 = 332 * sqrt(297.95/273.) m/s
V24.8 = 346.84 m/s

We Can clearly see the two values are nearly approximate.

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