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How many hydrogen atoms are in $35.0$ grams of hydrogen gas?

How many hydrogen atoms are in $35.0$ grams of hydrogen gas?

In order to understand the quantity of atoms in a given mass of element, we need to understand the concept of Mole. 

$Mole$ is defined as the mass of substance which can be an atom, molecule, electron, ion or any other particle or group of particles having $6.022\times{10}^{23}$  elementary entities which is known as $Avogadro’s$ $Constant$ or $Avogadro’s$ $Number$ having a symbol of $N_A$ which is expressed in SI unit ${\rm mol}^{-1}$. Mole is the $SI$ unit for the amount of substance which is represented by the symbol $mol$.

\[Avogadro’s Number = \frac{6.022\times{10}^{23}\ atoms}{1\ mol}\ \]

Mole is also similar to the substance’s atomic or molecular mass as examples given below:

  • Carbon has an atomic mass of $12$, hence $1$ $mol$ of atomic carbon will have a mass of $12$ $grams$ and contains $6.022\times{10}^{23}$ of carbon atoms.
  • Hydrogen has an atomic mass of $1.0079$, hence $1$ $mol$ of atomic hydrogen will have a mass of $1.00784$ $grams$ and contains $6.022\times{10}^{23}$ of Hydrogen atoms.
  • Water $H_2O$ has a molecular mass of $18.01528$, hence $1$ $mol$ of molecular water will have a mass of $18.01528$ $grams$ and contains $6.022\times{10}^{23}$ of water molecules.

 

Expert Answer:

We know that Molar mass of $H_2$ is equal to molecular mass of $H_2$. We will divide the given mass of element with molar mass of $H_2$ to get the number of moles. This is called conversion of given mas to number of moles

\[Mass\ \rightarrow\ Moles\]

Once you get the number of moles, multiply it with Avogadro’s Number to calculate the number of atoms. This is called conversion of number of moles to number of atoms.

\[Mass\ \rightarrow\ Moles\ \rightarrow\ Atoms\]

As per the concept of mole

\[\frac{m}{M}\ =\ \frac{N}{N_A}\]

Where,

$m =$ Mass of hydrogen gas $H_2 = 35g$

$M =$ Molar Mass of Hydrogen Gas $H_2 = 2.01568 \dfrac{g}{mol}$

$N_A =$ Avogadro’s Number  $= 6.022\times{10}^{23}$

$N =$ Number of Hydrogen $H_2$ Atoms

By re-arranging the equation and substituting the values, we get

\[N\ =\ \frac{35g}{2.01568\ \dfrac{g}{mol}}\ \times\ 6.022\times{10}^{23}{\mathrm{mol}}^{-1}\ \]

By cancelling out units of gram and mol,

\[N\ =\ 104.565\ \times\ {10}^{23}\]

By moving the decimal to two points left,

\[N\ =\ 1.04565\ \times\ {10}^{25}\]

Numerical Results:

As per the mole concept, the number of hydrogen atoms in $35g$ of Hydrogen Gas are $1.04565\ \times\ {10}^{25}$

Example:

Question: How many atoms of gold are in $58.27 g$ of gold $Au$ ?

We know that the atomic weight of gold, $Au$ is $196.967$.

So, the Molar Mass $M$ of Gold, $Au = 196.967 \dfrac{g}{mol}$

As per the concept of mole

\[\frac{m}{M}\ =\ \frac{N}{N_A}\]

Where,

$m =$ Mass of Gold $Au = 58.27g$

$M =$ Molar Mass of Gold $Au = 196.967 \dfrac{g}{mol}$

$N_A =$ Avogadro’s Number  $= 6.022\times{10}^{23}$

$N =$ Number of Gold $Au$ Atoms

By re-arranging the equation and substituting the values, we get

\[N\ =\ \frac{58.27g}{196.967\ \dfrac{g}{mol}}\ \times\ 6.022\times{10}^{23}{\mathrm{mol}}^{-1}\ \]

By cancelling out units of gram and mol, we get number of atoms of Gold as follows:

\[N\ =\ 1.782\ \times\ {10}^{23}\]

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