Name: Expression for Apparent Frequency
Uploaded: 2019-07-19
Duration: 6 min 59 s
Description: After understanding the basics of the Doppler effect, let us dive into mathematical expressions relating to the Doppler effect. Watch the video to find out.

Question

The atomic mass of iodine is $127 g {mol}^{- 1}$ . A standing wave in iodine vapour at $400 K$ has nodes that are $6.77 cm$ apart when the frequency is $1000 Hz$ . At this temperature, is iodine vapour monoatomic or diatomic?

Accepted Answer

We need to determine the atomicity of iodine molecule based on the properties of sound wave in its vapour.

Atomicity affects the molecular weight and that, in turn, affects the speed of sound.

Hence, we need to calculate the speed of sound and then infer the atomicity from this.

Given for iodine vapour,

temperature $T = 400 K$ ,
Atomic mass is $127 g {mol}^{- 1}$ .

Standing wave has,

frequency $f = 1000 Hz$
Distance between nodes $L = 6.77 cm = 6.77 \times 10^{- 2} m$ apart.

Now, wavelength $λ = 2 L$

From the above information, we can calculate speed of sound.

$v = f λ = 1000 \times 2 \times 6.77 \times 10^{- 2} = 135.4 m s^{- 1}$

Also, speed of sound in a vapour,

$v = \sqrt{\frac{γ R T}{M}} \Rightarrow M = \frac{γ R T}{v^{2}} . . . (1)$

where, $γ$ is the ratio of constant pressure and constant volume molar specific heat capacities.
$R (= 8.31 J {mol}^{- 1} K^{- 1})$ is the universal gas constant and $M$ is the molecular weight.

We also know that, $γ = \frac{5}{3}$ for monatomic gas and $\frac{7}{5}$ for diatomic gas.

We need to determine if iodine is monatomic or diatomic, or if the molecular weight is $127 \times 1 = 127$ or $127 \times 2 = 254$

For $γ = \frac{5}{3}$ , substituting values in equation $(1)$ ,

$M = \frac{\frac{5}{3} \times 8.31 \times 400}{135 . 4^{2}} = 0.3 kg {mole}^{- 1} = 300 g {mole}^{- 1}$

for $γ = \frac{7}{5}$ , we get $M = 0.254 kg {mole}^{- 1} = 254 g {mole}^{- 1}$

This matches with the value of molecular weight of diatomic iodine. Hence, we can conclude that iodine is a diatomic molecule.

The atomic mass of iodine is $127 g {mol}^{- 1}$ . A standing wave in iodine vapour at $400 K$ has nodes that are $6.77 cm$ apart when the frequency is $1000 Hz$ . At this temperature, is iodine vapour monoatomic or diatomic?

Important Questions on Sound Waves

Learn and Prepare for any exam you want !

The atomic mass of iodine is 127 g mol-1. A standing wave in iodine vapour at 400 K has nodes that are 6.77 cm apart when the frequency is 1000 Hz. At this temperature, is iodine vapour monoatomic or diatomic?

Important Questions on Sound Waves

Learn and Prepare for any exam you want !

The atomic mass of iodine is $127 g {mol}^{- 1}$ . A standing wave in iodine vapour at $400 K$ has nodes that are $6.77 cm$ apart when the frequency is $1000 Hz$ . At this temperature, is iodine vapour monoatomic or diatomic?