Question

The total charge in coulombs, of all the electrons in a three-gram mole hydrogen atom, is equal to \[ - x \times {10^3}C\] . Find the value of \[x\] .
Use Avogadro number, \[{N_A} = 6 \times {10^{23}}\] and charge on an electron \[e = - 1.6 \times {10^{ - 19}}C\] .

Answer 1

Hint: 3 gram-mole of hydrogen atoms means 3 moles of hydrogen atoms. The number of hydrogen atoms \[ = \] Number of moles of hydrogen atom \[ \times \] Avogadro’s constant. The total number of electrons in hydrogen atoms \[ = \] Number of hydrogen atoms \[ \times 1\] . Now finally calculate the total charge of all the electrons in three-gram mole hydrogen \[ = \] Total number of electrons \[ \times 1.6 \times {10^{ - 19}}\] .

Complete answer:
To solve this problem we have to first look at what gram-mole means
So 1 gram-mole in the simplest way is the weight of a substance in grams that is equal to the molar mass of that substance. Let’s take an example, consider we are given 3-gram moles of \[{O_2}\] molecule. 1 gram-mole of \[{O_2}\] molecules will weigh 16 grams (Molar mass of \[{O_2}\] is 16amu). So, 3-gram moles of \[{O_2}\] molecules will weigh 48 grams. Now, the important thing for us to note is the number of moles of \[{O_2}\] molecules in 3 gram-moles of \[{O_2}\] molecules.
So, \[n = \dfrac{{Mass}}{{Molar{\text{ }}Mass}}\]
\[n = \] Moles of \[{O_2}\] molecules
\[Mass = \] Weight of \[{O_2}\] molecules in grams
\[Molar{\text{ }}Mass = \] The molar mass of \[{O_2}\] molecules
So for \[{O_2}\] molecules
\[n = \dfrac{{48}}{{16}} = 3\] moles.
This means 1 mole-gram of a substance contains one mole of that substance.
Now, coming back to hydrogen atoms, we are given 3 mole-grams which means 3 moles of the hydrogen atom.
The number of hydrogen atoms \[ = \] moles \[ \times \] Avogadro’s constant
Number of hydrogen atoms \[ = 3 \times 6 \times {10^{23}}\]
Number of hydrogen atoms \[ = 18 \times {10^{23}}\] atoms
Now, we also know that 1 hydrogen atom is the simplest and consists of 1 electron, 1 proton, and no neutrons.
So, the number of total electrons in \[18 \times {10^{23}}\] hydrogen atoms will be
Total number of electrons \[ = 18 \times {10^{23}} \times 1\]
Total number of electrons \[ = 18 \times {10^{23}}\] electrons
Now, we know that charge of 1 electron is \[ - 1.6 \times {10^{ - 19}}C\]
So, the total charge in electrons of 3 gram-mole hydrogen \[ = 18 \times {10^{23}} \times 1.6 \times {10^{ - 19}} = - 2.88 \times {10^5}C\]
We are given the question that the charge on electrons in 3 gram-mole hydrogen is \[ - x \times {10^3}C\] .
So, \[ - x \times {10^3} = - 2.88 \times {10^5}\]
\[x = 288\]

Note:
In the solution, we discussed what gram-mole means by using the example of \[{O_2}\] molecule, but you can skip this step and just mention that 1 gram-mole of a substance contains 3 moles of a substance and the weight of 1 mole-gram of a substance will be equal to the molar mass of the substance in grams.