A student is performing an experiment using a resonance column and a tuning fork of frequency `244s^(-1)`. He is told that the air in the tube has been replaced by another gas (assuming that the air column remains filled with the gas). If the minimum height at which resonance occurs is `(0.350+- 0.005)m`, the gas in the tube is (Useful information : `sqrt(167RT) = 640J^(1//2)mol^(-1//2)`,
`sqrt(140RT) = 590J^(1//2)mol^(-1//2)`. The molar masses `M` in grams are given in the options. take the values of `sqrt((10)/(M))` for each gas as given there.)

Options

A. Neon `[M=20,sqrt((10)/(20))=(7)/(10)]`
B. Nitro g en `[M=28, sqrt((10)/(20))=(3)/(5)]`
C. Oxygen`[M=32, sqrt((10)/(32))=(9)/(16)]`
D. Argon`[M=36, sqrt((10)/(36))=(17)/(32)]`

Answer

D

Explanation

Let `l` be the length of resonance column. Then
`l=lambda//4 ` or `lambda=4l`
velocity, `upsilon=vlambda=v4l=244xx4xxl` …(i)
When `l=0.350+0.005=0.355m, ` then ` upsilon=244xx4xx0.355=346.5m//s`
When `l=0.350-0.005=0.345m,` then
`upsilon=244xx4xx0.345=336.5m//s`
Thus, the value of `upsilon` lies between `336.5m//s` to `346.5m//s`
We know that, `upsilon=sqrt((gammaP)/(rho))=sqrt((gammaPV)/(M))=sqrt((gammaRT)/(M))`
Here, `M` is molecular mass of gas. If it is in gram, then
`upsilon=sqrt((gammaRT)/(Mxx10^(-3)))=sqrt(100gammaRT)xxsqrt((10)/(M))`
For monoatomic gas, `gamma=1.67` and for diatomic gas` gamma=1.4`
`upsilon_(NE)sqrt(100xx1.67xxRT)xxsqrt((10)/(20))=640xx(7)/(10)`
`=448m//s`
`upsilon_(AR)=sqrt(100xx1.67xxRT)xxsqrt((10)/(36))=640xx(17)/(32)`
`=340m//s`
`upsilon_(O_(2))=sqrt(100xx1.4RT)xxsqrt((10)/(32))=590xx(9)/(16)`
`=331.8m//s`
`upsilon_(N_(2))=sqrt(100xx1.4RT)xxsqrt((10)/(28))=590xx(3)/(5)`
`=354 m//s. `
`:.` the only possible choice is Argon.