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

Hint: In this question we have been asked to determine whether the given iodine vapour is monatomic or diatomic. We know that this can be determined by calculating the value of adiabatic index. We have been given the atomic mass, the frequency and temperature of iodine vapour. Therefore, we shall use the equation based on kinetic theory of gases. It states the relation between given parameters and adiabatic index.
Formula Used: - \[v=\sqrt{\dfrac{\gamma RT}{M}}\]
v is the velocity of sound wave
R is the gas constant
T is temperature in kelvin
M is the molar mass
\[\gamma \] is the adiabatic index.

Complete step by step solution:
We know that, when the value of adiabatic index \[\gamma \] is 1.66 the vapour is said to be monatomic and when it is 1.44 the vapour is said to be diatomic.
Therefore, solving for adiabatic index \[\gamma \]
From the equation of kinetic theory of gases,
\[v=\sqrt{\dfrac{\gamma RT}{M}}\] ………………… (1)
We also know that,
\[v=\mu \lambda \] ……………….. (2)
It is frequency is 1000 Hz
We know wavelength \[\lambda \] is given by,
\[\lambda =2I\]
Where I is the distance between nodes that is given to 6.77 cm.
After substitution,
\[\lambda =2\times 6.77cm\]
On solving,
\[\lambda =0.135m\] …………….. (3)
From (2) and (3)
\[v=1000\times 0.135\]
Therefore,
\[v=135m/s\] …………………. (4)
Therefore, from (1) and (4)
We get,
\[135=\sqrt{\dfrac{\gamma RT}{M}}\]
It is given that atomic mass of the iodine vapour is
\[m=127g/mol\]
Therefore, molar mass is
\[M=nm=0.127n\] kg/mol
Therefore, after substituting all the values in above equation
We get,
\[135=\sqrt{\dfrac{\gamma \times 8.3\times 400}{0.127n}}\]
Therefore,
\[\gamma =0.7n\]
Now, for the value of n = 2 ………….. (since the molecular formula of iodine is \[{{I}_{2}}\])
we get,
\[\gamma =1.4\]
Therefore, the given atomic vapour is diatomic.

Note: The monatomic gases consist of a single atom. All the noble gases are an example of monoatomic elements. Similarly, diatomic gases consist of atoms. The diatomic elements can not exist solo i.e. the fluorine or nitrogen. The kinetic theory of gases states that a gas consists of molecules in random motion. These molecules influence each other only by collisions and all these collisions are perfectly elastic. The volume occupied by molecules is negligible.