# The molecular weight of a hydrogen molecule is M=2.016×${{10}^{-3}}$ kg/mol. If the root-mean-square speed of hydrogen molecules (H$_{2}$ ​) at 373.15K(100${}^{\circ }C$) is ${v_{rms}}$​=$\dfrac{x+210}{100}km/s$ Find x.

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Hint: At first, we need to calculate the mass of hydrogen from the molecular weight of hydrogen that is given. Then compare the equation to the formula for kinetic energy. After getting the $V_{rms}$, put it in the given equation and solve it for the value of x.

The mass of hydrogen molecule can be calculated from molecular weight,
$m=\dfrac{M}{{{N}_{0}}}=\dfrac{2.016\times {{10}^{-3}}kg/mol}{6.02\times {{10}^{23}}mo{{l}^{-1}}}=3.35\times {{10}^{-27}}kg$
Then,
$\dfrac{1}{2}mv_{rms}^{2}=\dfrac{3}{2}kT$
On solving,
$\Rightarrow {{v}_{rms}}=\sqrt{\dfrac{3kT}{m}}$
$\Rightarrow {{v}_{rms}}=\sqrt{\dfrac{3\times 1.38\times {{10}^{-23}}\times 373.15}{3.35\times {{10}^{-27}}}}$
$\Rightarrow {{v}_{rms}}=2.15Km/s$
Now,
${{V}_{rms}}=\dfrac{x+210}{100}km/s$
2.15=$\dfrac{x+210}{100}$,
which on solving, we get the value of x as 5 km/s

The ${{V}_{rms}}$ is defined as the square root of the mean of the squares for one time period of the sine wave.