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State Newton’s formula for the velocity of sound in a gas. What is Laplace’s correction?

Answer
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Hint:
-Newton formulated the velocity of sound in gas by assuming that the sound wave going through the gas medium in the isothermal process. That means in the time of propagation of the sound the pressure and the volume of many parts of the gas medium can be changed due to the compression and rarefaction but the temperature of the medium remains constant.
-Laplace corrected the assumption of Newton by telling that in the time of propagation of the sound the temperature can not be constant. So, the propagation of sound in the gas medium is an adiabatic process.

Complete step by step answer:
Newton’s Calculation: The velocity of sound that propagates in a gas medium was first formulated by Newton.
He stated the formula of the velocity of the propagation of sound in a gas medium by assuming that the sound wave going through the gas medium in the isothermal process.
In the time of propagation of the sound, the pressure and the volume of many parts of the gas medium can be changed due to the compression and rarefaction but the temperature of the medium remains constant.
Let us consider, for a part of the gas a certain amount of pressure is P and volume is V. During the propagation of the sound, the pressure becomes (P+P1) by increasing and hence the volume becomes (Vv) by decreasing.
Since the temperature is constant, from Boyle's law,
PV=(P+P1)(Vv)
On multiplying the term and we get,
PV=PV+P1VPvP1v
0=P1VPv [The term P1v is too small and hence neglected]
Pv=P1V
P1Vv=P
On rewriting the terms and we get,
P=P1vV=volume stressvolume strain
bulk modulus = k=volume stressvolume strain
P=k
We know, if the sound propagating in a gas medium of density ρ and bulk modulus k, the velocity of sound is,
c=kρ
So, from Newton’s calculation, we get, c=Pρ.

Laplace’s Correction: Laplace first told that during the propagation of sound in the gas medium the temperature does not constant. The assumption of Newton of the isothermal process is not right.
Laplace assumed that during the compression and rarefaction the expansion and compression of the layers are so fast to keep the temperature constant. Hence the process should be the adiabatic process instead of the isothermal process.
In the adiabatic process, if a part of the gas has a certain amount of pressure is P and volume is V,
PVγ=constant.
Hence, During the propagation of the sound, if the pressure becomes (P+P1) by increasing and hence the volume becomes (Vv) by decreasing.
PVγ=(P+P1)(Vv)γ
PVγ=(P+P1)Vγ(1vV)γ
P=(P+P1)(1γvV) [ for Binomial expansion the other terms are neglected]
P=P+P1PγvVP1γvV
P1=γPvV [The term P1v is too small and hence neglected]
γP=P1vV=volume stressvolume strain
bulk modulus = k=volume stressvolume strain
γP=k
So, from Laplace's Correction we get, c=γPρ.

Note:
-From Newton’s calculation, we get, c=Pρ.
Now for STP, the value of the velocity will be c=76×13.6×9800.00129328000cm/s280m/s. But the value obtained from many experiments is 332m/s.

-From Laplace's Correction we get, c=γPρ.
Now for STP, the value of the velocity will be c=1.4×76×13.6×9800.001293=33117cm/s=331.17m/s. This value is very much nearer to the experimental value(i.e. 332m/s).

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