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The molar mass of iodine is 127 g/mol. Hen sound at frequency 1000 Hz is introduced to a tube of iodine gas at 400 K, an internal acoustic standing wave is set up with nodes separated by 9.57 cm. What is $\gamma$ for the gas?

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Last updated date: 13th Jun 2024
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Answer
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Hint: The distance between two consecutive nodes and an antinode in a standing wave is represented as the half of the wavelength of the waves produced.
Distance = $\dfrac{\lambda}{2}$
Where lambda is the wavelength of the wave produced.

Complete step by step solution:
Mass of iodine = 127g/mol
T=400K
Distance between two node is= \[\dfrac{\lambda }{2} = 9.57 \times {10^{ - 2}}\]
\[\Rightarrow \dfrac{\lambda }{2} = 9.57 \times {10^{ - 2}}\\ \Rightarrow \lambda = 19.04 \times {10^{ - 2}}\\ \Rightarrow {{\rm{V}}_{{\rm{sound}}}} = f\lambda \\ \Rightarrow {{\rm{V}}_{{\rm{sound}}}} = 1000 \times 19.04 \times {10^{ - 2}}\\\therefore {{\rm{V}}_{{\rm{sound}}}} = 190.4m/\sec \\{\rm{also V = }}\sqrt {\dfrac{{{\rm{\gamma RT}}}}{{\rm{M}}}} \\ \Rightarrow \sqrt {\dfrac{{{\rm{\gamma RT}}}}{{\rm{M}}}} = 190.4\\{{\rm{V}}_{{\rm{sound}}}}{\rm{ = }}\dfrac{{{{{\rm{(190}}{\rm{.4)}}}^{\rm{2}}}{\rm{ \times 127 \times 1}}{{\rm{0}}^{{\rm{ - 3}}}}}}{{{\rm{8}}{\rm{.31 \times 400}}}}\\{\rm{\gamma = 0}}{\rm{.7}}\\\\\]

Additional Information:
Nodes and antinodes are known to form static waves. In given static waves, the distance between any two given continuous nodes is half the wavelength. The approximate distance between the node and the immediate next antinode is actually a quarter of the given wavelength.
In other words, the total distance or difference between two continuous nodes and an antinode in a given current waveform is usually represented as half the length of the wave produced.
The formula is Distance= $\dfrac{\lambda}{2}$
Where lambda is the wavelength. It is also to be known that the standing wave varies in distance proportionately to its wavelength with the displacement being zero always

Note: Nodes and antinodes are formed in static waves. In static waves, the distance between two continuous nodes (anti-nodes) is one half wavelength. Thus, the distance between a node and the immediate next antinode is a quarter of a wavelength. In the standing wave distance varies according to wavelength but the displacement is always zero.