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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?

Last updated date: 12th Sep 2024
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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= $\dfrac{\lambda }{2} = 9.57 \times {10^{ - 2}}$
$\Rightarrow \dfrac{\lambda }{2} = 9.57 \times {10^{ - 2}}\\ \Rightarrow \lambda = 19.04 \times {10^{ - 2}}\\ \Rightarrow {{\rm{V}}_{{\rm{sound}}}} = f\lambda \\ \Rightarrow {{\rm{V}}_{{\rm{sound}}}} = 1000 \times 19.04 \times {10^{ - 2}}\\\therefore {{\rm{V}}_{{\rm{sound}}}} = 190.4m/\sec \\{\rm{also V = }}\sqrt {\dfrac{{{\rm{\gamma RT}}}}{{\rm{M}}}} \\ \Rightarrow \sqrt {\dfrac{{{\rm{\gamma RT}}}}{{\rm{M}}}} = 190.4\\{{\rm{V}}_{{\rm{sound}}}}{\rm{ = }}\dfrac{{{{{\rm{(190}}{\rm{.4)}}}^{\rm{2}}}{\rm{ \times 127 \times 1}}{{\rm{0}}^{{\rm{ - 3}}}}}}{{{\rm{8}}{\rm{.31 \times 400}}}}\\{\rm{\gamma = 0}}{\rm{.7}}\\\\$

The formula is Distance= $\dfrac{\lambda}{2}$