To find the distance d over which a signal can be seen clearly in foggy conditions- a railway-s engineer uses dimensional analysis and assumes that the distance depends on the mass density - of the fog- intensity power-area S of the light from the signal and it-s frequency f- The engineer finds that d is proportional to S 1 - n- The value of n is

According to question d ∝ρ^a S^1/n f^b nbsp; nbsp; nbsp; nbsp; nbsp;... (1) Using the principle of dimensional homogeneity in equation (1) L^1= [ ...

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Byju's Answer

Standard XII

Chemistry

Compressibility Factor

To find the d...

Question

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railway's engineer uses dimensional analysis and assumes that the distance depends on the mass density $ρ$ of the fog, intensity (power/area) $S$ of the light from the signal and it's frequency $f$ . The engineer finds that $d$ is proportional to $S^{1 / n}$ . The value of $n$ is

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Solution

According to question
$d \propto ρ^{a} S^{1 / n} f^{b}$ ... (1)
Using the principle of dimensional homogeneity in equation (1)
$L^{1} = {[M^{1} L^{- 3}]}^{a} {[\frac{M^{1} L^{2} T^{- 3}}{L^{2}}]}^{1 / n} {[T^{- 1}]}^{b}$
$L^{1} = M^{a + \frac{1}{n}} L^{- 3 a} T^{\frac{- 3}{n} - b}$
On equating powers, we get
$- 3 a = 1 \Rightarrow a = - 1 / 3$
and $a + \frac{1}{n} = 0$
$\Rightarrow \frac{- 1}{3} + \frac{1}{n} = 0$
$\Rightarrow n = 3$

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According to question d∝ρaS1/nfb ... (1) Using the principle of dimensional homogeneity in equation (1) L1=[M1L−3]a[M1L2T−3L2]1/n[T−1]b L1=Ma+1nL−3aT−3n−b On equating powers, we get −3a=1⇒a=−1/3 and a+1n=0 ⇒−13+1n=0 ⇒n=3