Question

When a narrow resonance column apparatus is filled with ozone gas ($\gamma =4/3)$, it resonates in 1st resonance with a tuning fork. When it is filled with nitrogen gas $N_2$, it resonates with the same tuning fork in 1st resonance when water column is shifted down by $10cm$. Neglect end correction and take $\sqrt{5}=2.25$.

• A. Wavelength of sound in ozone is approximately $1m$
• B. Wavelength of sound in nitrogen is approximately $1.6m$
• C. For second resonance in ozone gas, the water column should be shifted by $60\text{cm}$
• D. For second resonance in nitrogen gas, the water column should be shifted by $80\text{cm}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B Wavelength of sound in nitrogen is approximately $1.6m$
C For second resonance in ozone gas, the water column should be shifted by $60\text{cm}$
D For second resonance in nitrogen gas, the water column should be shifted by $80\text{cm}$

Fundamental frequency of an air column closed at one end
For ozone :
$n_{1_{O_3}}=\frac{V_{O_3}}{4L}=\frac{1}{4L}\sqrt{\frac{{\gamma }_{O_3}RT}{M_{O_3}}}$

For nitrogen :
$n_{1_{N_2}}=\frac{V_{N_2}}{4(L+10)}=\frac{1}{4(L+10)}\sqrt{\frac{{\gamma }_{N_2}RT}{M_{N_2}}}$



$\frac{n_{1_{0_3}}}{n_{1_{N_2}}}=\frac{L+10}{L}\sqrt{\frac{4\times 5\times 28}{3\times 7\times 48}}$
$\Rightarrow \frac{L+10}{L}=\frac{3}{2.25}\Rightarrow 2.25L+22.5=3L$

$\Rightarrow 0.75L=22.5$
$\Rightarrow L=\frac{22.5}{0.75}=30\text{cm}$

$a)$ Wavelength of sound in $O_3$ ${\lambda }_{0_3}=4L=1.2\text{m}$


$b)$ Wavelength of sound in $N_2$ ${\lambda }_{N_2}=4(L+10)=\text{1.6 m}$


$c)L_1=\text{30 cm}$
$\Rightarrow$ Length of water column for second resonance $L_2=3L_1=90\text{cm}\Rightarrow L_2-L_1=60\text{cm}\text{(For}{O}_3)$

$d)L_1=\text{40 cm}$
$\Rightarrow$ Length of water column for second resonance $L_2=3L_1=120\text{cm}\Rightarrow L_2-L_1=80\text{cm}\left(\text{For}{N}_2\right)$