Question

The numerical value of N/n (where ‘N’ is the number of molecules in a given sample of gas and ‘n’ is the number of moles of the gas) is:
\[
  A.\;\;\;\;\;8.314\; \\
  B.\;\;\;\;\;6.02 \times {10^{23}} \\
  C.\;\;\;\;\;0.0821 \\
  D.\;\;\;\;\;1.62 \times {10^{ - 24}} \\
 \]

Answer 1

Hint:
The number of atoms in one mole of any substance is equal to Avogadro’s Number i.e. \[6.02 \times {10^{23}}\]atoms per mole
Formula used:\[N = {N_A} \times n\]

Complete step by step answer:
-We can quantify matter on the basis of its weight into various forms of measures to quantify the elementary particles present within the given sample of matter.
-A mole represents a very large number of units i.e. \[6.02 \times {10^{23}}\] units. Now you may be wondering why mole is considered as a mode of measurement of the amount of matter. The simple explanation to this is that experimentally, it has been found out that 1mol of any substance contains the same number of molecules of the given sample of matter. One mole of a substance is equivalent to the sample of the given substance if its weight is equal to the molecular or atomic weight of the substance. The number of molecules in one mole of any substance is equal to Avogadro’s Number i.e. atoms \[6.02 \times {10^{23}}\]molecules per mole.
-The relation between the number of molecules in a given sample and the number of moles of the same sample is directly proportional. Also, by simple mathematics, we can understand that if 1 mol of a substance contains \[6.02 \times {10^{23}}\]molecules, then ‘n’ mols of the substance will contain \[n \times 6.02 \times {10^{23}}\]molecules.
-If we write Avogadro's Number as \[{N_A}\], then we can say that the number of molecules present in ‘n’ moles of the substance is \[{N_A} \times n\].
Hence, the final representation of these relations can be shown as:
                                                    \[N = {N_A} \times n\]
Dividing both sides by ‘n’
\[\therefore \dfrac{N}{n} = \dfrac{{{N_A} \times n}}{n}\]
\[\therefore \dfrac{N}{n} = {N_A}\]
\[\therefore \dfrac{N}{n} = 6.022 \times {10^{23}}\]

Hence, Option B is the correct answer.

Note:

The value of N/n is universal in nature. This means that the variation in either the elemental gas or the number of moles of it present would not be affected. Hence, N/n is a constant value and is the same for all elements.