Question

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 $v_{rms}$​=$${\displaystyle \frac{x+210}{100}}km/s$$ Find x.

Answer 1

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.

Complete step by step answer:
The mass of hydrogen molecule can be calculated from molecular weight,
$m={\displaystyle \frac{M}{N_0}}={\displaystyle \frac{2.016\times {10}^{-3}kg/mol}{6.02\times {10}^{23}mo{l}^{-1}}}=3.35\times {10}^{-27}kg$
Then,
${\displaystyle \frac{1}{2}}m{v}_{rms}^2={\displaystyle \frac{3}{2}}kT$
On solving,
$\Rightarrow v_{rms}=\sqrt{{\displaystyle \frac{3kT}{m}}}$
$\Rightarrow v_{rms}=\sqrt{{\displaystyle \frac{3\times 1.38\times {10}^{-23}\times 373.15}{3.35\times {10}^{-27}}}}$
$\Rightarrow v_{rms}=2.15Km/s$
Now,
$V_{rms}={\displaystyle \frac{x+210}{100}}km/s$
2.15=${\displaystyle \frac{x+210}{100}}$,
which on solving, we get the value of x as 5 km/s

Additional Information:
The sum of the atomic masses of all atoms in a molecule, based on a scale in which the atomic masses of different elements is known as Molecular weight. For example, the molecular weight of water, which has two parts of hydrogen and one part of oxygen, is 18 (i.e., 2 + 16).
The $V_{rms}$ is defined as the square root of the mean of the squares for one time period of the sine wave.
A molecule of hydrogen is the simplest possible molecule. It consists of two protons and two electrons that are held together by forces. Like atomic hydrogen, the assemblage can exist in one or more energy levels.
The root mean square (RMS or rms) is defined as the square root of the mean square. The RMS is also known as the quadratic mean and is a particular case of where the generalized mean is with exponent 2.

Note:
Students must calculate the mass of the hydrogen molecule from molecular weight properly. Relation between rms value and kinetic energy must be clear. Always remember that hydrogen is a diatomic molecule.