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Calculate the root mean square velocity of the hydrogen molecule at STP.

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Last updated date: 20th Jun 2024
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Answer
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Hint: Recall the formula used to calculate the root mean square velocity. Also take into consideration the kinetic molecular orbital theory. Think about what values you can put in the formula given that the molecules are present at STP.

Complete step by step answer:
Remember that STP means Standard Temperature Pressure. According to this, we will solve the problem.
We know that according to the Kinetic Molecular Theory of Gases, kinetic energy of molecules is defined as the product of the universal gas constant (R) and the ambient temperature (T) multiplied by the constant 3/2. So, the relation is defined as:
\[{{E}_{k}}=\frac{3}{2}RT\]
From classical mechanics, we know that kinetic energy is defined as half into the product of the mass and the square of the velocity. The equation we get is:
\[{{E}_{k}}=\frac{1}{2}m{{v}^{2}}\]
Now equating both these equations, we get:
\[\frac{3}{2}RT=\frac{1}{2}m{{v}^{2}}\]
The velocity that is referred to here is considered to be the mean velocity of all the molecules present in the given volume. We will consider the mass (m) to be the molar mass of the given substance since we have to find the velocity of hydrogen molecules at STP; which means that 1 mole of hydrogen gas molecules is present, so we will denote $m$ as $M$ which is the molar mass of the hydrogen molecules. Now, we will solve the equation for $v$ by rearranging it and taking the root.
\[\begin{align}
  & {{v}^{2}}=\frac{3}{2}\times \frac{2}{1}\times \frac{RT}{M} \\
 & {{v}^{2}}=\frac{3RT}{M} \\
 & {{v}_{rms}}=\sqrt{\frac{3RT}{M}} \\
\end{align}\]
Thus, we have the formula for the root mean square velocity. Now, let us figure out the values that we have to put in to get the root mean square velocity of hydrogen molecules at STP.
We know that the temperature at STP is $0{}^\circ C$. But we need all the values to be in SI units. So, we will convert the value given into Kelvins. So, the temperature at STP will be $273.15K$.
The molar mass of the hydrogen gas molecules has to be defined in kilograms. We know that according to the mole concept, 1 mole of any substance has a mass that is equivalent to the molecular mass of that substance in grams. The number of atoms of hydrogen present in one molecule of hydrogen is 2. So, we will multiply the mass of 1 atom of hydrogen of 2.
Molecular mass of ${{H}_{2}} =2\times 1amu$
Mass of 1 mole of ${{H}_{2}}=2grams$
Now, we will convert this value into kilograms.
Molar mass of ${{H}_{2}}=2\times {{10}^{-3}}kg$
We will move to the universal gas constant now that we know the units of the other variables. Since, we are considering all the units to be SI units, the value of the universal gas constant will be:
\[R=8.314Jmo{{l}^{-1}}{{K}^{-1}}\]
 Now, putting all these values in the formula for the root mean square velocity, we get:
\[{{v}_{rms}}=\sqrt{\frac{3\times 8.314\times 273.15}{2\times {{10}^{-3}}}}\]
Now, solving for ${{v}_{rms}}$ we get:
\[\begin{align}
  & {{v}_{rms}}=\sqrt{3406.454\times {{10}^{3}}} \\
 & {{v}_{rms}}=\sqrt{3.406\times {{10}^{6}}} \\
\end{align}\]
Taking the square root, we get:
\[{{v}_{rms}}=1.845\times {{10}^{3}}\]
Hence, the root mean square velocity of hydrogen molecules at STP is $1.845\times {{10}^{3}}m{{s}^{-1}}$.

Additional Information: We can solve this problem using another method. Use the formula $PV=nRT$ to substitute $RT$ by $\frac{PV}{n}$ and take the values of pressure and volume at STP (take pressure in pascals and volume in ${{m}^{3}}$).

Note: Remember to convert all the units to SI units to simplify the calculations while solving the equation. You can consider the units as convenient for you but make sure that you take the value of the universal gas constant according to those units.