Question

When a narrow resonance column apparatus is filled with ozone gas $(y = \dfrac{4}{3})$, it resonates in $1st$ resonance with a tuning fork. When it is filled with nitrogen gas ($N2$), it resonates with the same tuning fork in $1st$ resonance when the water column is shifted down by $10$cm. (Take $V5 = 2.25$)
A. wavelength of sound in ozone is approximately $1$m.
B. wavelength of sound in Nitrogen is approximately $1.6$m.
C. For second resonance in ozone gas, the water column should be shifted down by $60$cm.
D. For second resonance in Nitrogen gas, the water column should be shifted by $80$cm.

Answer 1

Hint: Here we will use the formula for the speed of sound, the standard formula and accordingly will find the ratios of two different speeds. Then will check for the given multiple choices are applicable or not.

Complete step by step answer:
Speed of Sound can be given by
Speed $ = \sqrt {\dfrac{{rRT}}{M}} $
When there are two different speeds, the ratio of both the speed can be given as-
$\dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{{{r_1}{M_2}}}{{{r_2}{M_1}}}} $
Place the given values in the above equation-
$\dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{{4 \times 5 \times 28}}{{3 \times 7 \times 40}}} $
Simplify the above expression-
$\dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{2}{3}} $
Now, the ratios of the wavelength can be given as –
$\dfrac{{{\lambda _1}}}{{{\lambda _2}}} = \dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{2}{3}} $
Also, we have ${\lambda _2} = {\lambda _1} + 10$
Place in the above equation –
$\dfrac{{{\lambda _1}}}{{{\lambda _1} + 10}} = \sqrt {\dfrac{2}{3}} $
Simplify the above equation –
$\dfrac{{{\lambda _1}}}{{{\lambda _1} + 10}} = 0.8$
Make the required unknown the subject –
$
  0.2{\lambda _1} = 8 \\
  {\lambda _1} = 40cm \\
 $
This is the required solution.

So, the correct answer is Option B,CD.

Note: Always remember that the numerator’s denominator goes in denominator and the denominator’s denominator goes to the numerator of the fraction. Be good in simplification of basic mathematical concepts. Always remember when we take a ratio of two speeds, then the constant terms of both the speeds are removed since they cancel each other.