Question

Find (a) the total number of neutrons and (b) the total mass of neutrons in \[7mg\] of \[^{14}C\].
[Assuming that mass of a neutron = \[1.675 \times {10^{ - 27}}kg\]].

Answer 1

Hint: An atom comprises a nucleus having neutrons and protons which are surrounded by electrons in orbits. Number of protons (atomic number) can be considered to be equal to the total number of electrons in a neutral atom (denoted as Z). Whereas, the summation of the number of neutrons and number of protons equals the mass number of an atom (denoted as M). Thus, the number of neutrons can be calculated by subtracting the atomic number (Z) from the mass number of an atom (M).

Complete step by step answer:
Molar mass of\[^{14}C{\text{ }} = {\text{ }}14g\]. Thus, first of all, we will estimate the number of moles in \[7mg\](i.e. \[0.007{\text{ }}g\]) of \[^{14}C\].
${\text{Number of moles = }}\dfrac{{{\text{mass}}}}{{{\text{molar mass}}}}$\[\]


Thus, number of moles in \[0.007{\text{ }}g\] of \[^{14}C\]:
$ = \dfrac{{0.007}}{{14}} = 0.0005{\text{ moles}}$

We know that, ${\text{1 mole of}}{{\text{ }}^{14}}C = 6.023 \times {10^{23}}{\text{ atoms of}}{{\text{ }}^{14}}C$

Thus, number of atoms in 0.0005 moles of 14C = $0.0005 \times 6.023 \times {10^{23}}{\text{ atoms of}}{{\text{ }}^{14}}C$
$ = 3.015 \times {10^{20}}{\text{atoms of}}{{\text{ }}^{14}}C$

Now, we know that atomic number of carbon = 7. So 1 atom of \[^{14}C\] comprises 6 protons and 8 neutrons. Therefore, the total number of neutrons in 7 mg of \[^{14}C\] can be calculated as follows:
As already mentioned, 1 atom of \[^{14}C\] = 8 neutrons:
$3.015 \times {10^{20}}{\text{atoms of}}{{\text{ }}^{14}}C = 8 \times 3.015 \times {10^{20}}{\text{neutrons of}}{{\text{ }}^{14}}C$
${\text{ = 2}}{\text{.412}} \times {\text{1}}{{\text{0}}^{21}}{\text{neutrons of}}{{\text{ }}^{14}}C$
The mass of 1 neutron = \[1.675 \times {10^{ - 27}}kg\] (Given)

Therefore, the mass of \[2.412 \times {10^{21}}\]neutrons of \[^{14}C\] can be calculated as stated below:
\[
  {\text{2}}{\text{.412}} \times {\text{1}}{{\text{0}}^{21}} \times 1.675 \times {10^{ - 27}}
   = 4.04 \times {10^{ - 6}}kg
 \]
 As a result, (a) the total number of neutrons in \[7mg\] of \[^{14}C\] ${\text{ = 2}}{\text{.412}} \times {\text{1}}{{\text{0}}^{21}}{\text{neutrons of}}{{\text{ }}^{14}}C$ and
(b) the total mass of neutrons in \[7mg\] of \[^{14}C\] \[ = 4.04 \times {10^{ - 6}}kg\].

Note: Each atom of a given element has the similar number of protons while atoms of distinct elements possess distinct numbers of protons. An atom consists of the same number of protons as well as electrons. As protons as well as electrons possess equal and opposite electrical charges, atoms have no overall electrical charge (i.e. neutral). For instance, the atomic number of sodium equals 11. This means that each sodium atom possesses 11 protons and 11 electrons.