Discrepancy of Newton’s Formula and Laplace Correction

The discrepancy between the theoretical value calculated by Newton and the experimental value is corrected by Laplace. He assumed that the process of formation of compression and rarefaction is very rapid process. He also argued that heat is neither lost during compression nor taken from the surrounding during rarefaction.

So, the system is adiabatic. For adiabatic process,

PV γ  = Constant……(i)

Where γ  is the ratio of molar heat capacity of air at constant pressure to that at constant temperature.  It’s average value is 1.4 for diatomic gas. Now, differentiating equation (i), we get:

$$ \text{d(PV)}^ \gamma) =  \text{d(Constant)}$$

$$\text{or, PV}^ \gamma  +  \gamma \text{PV}^{ \gamma -1}.\text{dV} = 0$$

$$\text{or, V}^ \gamma \left ( \text{dP} +  \frac{ \gamma \text{PdV}}{\text{V}} \right ) = 0$$

$$\text{or, dP} +  \gamma  \frac{\text{PdV}}{\text{V}} = 0$$

$$\gamma {\text{p}} = \left (  \frac{\text{dP}}{ \frac{\text{dV}}{\text{V}}}  \right ) =  {\text{B}}……(ii)$$

For speed of sound in gas medium,

$${\text{V}} =  \sqrt{\frac{ \beta }{ \rho}}$$

$${\text{V}} =  \sqrt {\frac{  \gamma P}{ \rho}}$$

This is the  speed of sound in gas medium corrected by Laplace.

At Normal temperature and pressure:

ρ = 1.295 kgm-3

T = 273 K

P = 1.013 × 105 Pa

γ = 1.4

Now, $ \text{V} = \sqrt{\frac{ \gamma \text{P}}{\rho}}$

$ \text{or,}  \text{V} = \sqrt{\frac{1.4 × 1.013 × 10^5}{1.293}}$

$ \therefore \text{V} = 331 ms^{-1}$

The value is very close to the experimental value (332ms-1). So, this value is accepted.

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