Question

If the diameter of a carbon atom is $0.15$ nm, calculate the number of carbon atoms which can be placed side by side in a straight line across a scale of length 20 cm long.

Answer 1

Hint: If atoms of equal diameters are placed side by side in a straight line, then the total length of the line will be equal to the sum of the diameters of each atom.
In other words, the total length of the line will be equal to the product of the number of atoms and the diameter of each atom.

Complete step by step answer:
-Given that, the diameter of a carbon atom is $0.15$ nm. Also given that the length of the straight line along which the carbon atoms are to be placed is 20 cm long.
-We need to find out the number of carbon atoms which can be placed side by side in the straight line of length 20 cm.
-Let, n be the number of carbon atoms to be placed side by side in the straight line of the given length of 20 cm. Then , the product of the number of carbon atoms ‘n’ and the diameter of each carbon atom will be equal to the length along which the carbon atoms are to be placed.
-If we convert the diameter of the carbon atom from nm to m, we get $0.15 \times {10^{ - 9}}$ m.
If we convert the length of the straight line from cm to m, we get $20 \times {10^{ - 2}}$ m.
Thus, we have:
$
  {\text{n}} \times {\text{0}}{\text{.15}} \times {10^{ - 9}} = 20 \times {10^{ - 2}} \\
   \Rightarrow {\text{n}} = \dfrac{{2 \times {{10}^{ - 1}}}}{{1.5 \times {{10}^{ - 10}}}} \\
   \Rightarrow {\text{n}} = 1.33 \times {10^9} \\
 $
Hence, the number of carbon atoms each of diameter $0.15$ nm which can be placed side by side in the straight line of length 20 cm is $1.33 \times {10^9}$ .

Note:
If radius of the carbon atoms is given instead of diameter, then the product of the number of carbon atoms ‘n’ and twice the radius of each carbon atom will be equal to the length along which the carbon atoms are to be placed because diameter is twice the radius.
Mathematically, if d is the diameter and r is the radius, then d is equal to 2r and the relation with the total length ‘L’ and the number of atoms ‘n’ can be expressed as:
${\text{n}} \times 2{\text{r}} = {\text{L}}$