Question

The dot at the end of this sentence has a mass of about one microgram. Assuming that the black stuff is carbon, the approximate number of atoms of carbon needed to make such a dot is $5 \times {10^x}$. The value of the dot is-

Answer 1

Hint:First; convert the unit from microgram to gram. Then use the formula of moles which is given as-
Moles=$\dfrac{{{\text{Given mass }}}}{{{\text{Atomic mass }}}}$ Where given mass is the mass of the dot and atomic mass is mass of one carbon atom. Put the values in the formula and solve it. Then multiply it by the Avogadro number to get the number of total carbon atoms needed to make the dot. Then compare the obtained value and the given value in the question to get the value of x.

Complete step-by-step answer:Given, the mass of the dot (m) =$1$ microgram
We know that, $1$ microgram=${10^{ - 6}}{\text{ g}}$
We have to find the value of x in $5 \times {10^x}$ that is the number of atoms of carbons needed to make the dot.
We know that the atomic mass of one carbon atom is=$12$ g/mol
We know the formula of moles=$\dfrac{{{\text{Given mass }}}}{{{\text{Atomic mass }}}}$
Then moles of carbon atom in one microgram=$\dfrac{{{\text{given mass of the dot}}}}{{{\text{atomic mass of carbon}}}}$
On putting the values in the formula, we get-
$ \Rightarrow $ Moles of carbon atom in one microgram=$\dfrac{{{{10}^{ - 6}}}}{{12}}{\text{ g}}{\text{.mol}}{{\text{g}}^{ - 1}}$
On solving, we get-
$ \Rightarrow $ Moles of carbon atom in one microgram=$0.0833 \times {10^{ - 6}}{\text{moles}}$-- (i)
Then the total number of carbon atoms needed to make the dot= Moles of carbon atom ×Avogadro number-- (ii)
We know that Avogadro number=$6.023 \times {10^{23}}$_{-- (iii)}
On putting the values of eq. (i) and eq. (iii) in eq. (ii), we get-
$ \Rightarrow $ The total number of carbon atoms needed to make the dot=$0.0833 \times {10^{ - 6}} \times 6.023 \times {10^{23}}$
On multiplication, we get-
$ \Rightarrow $ The total number of carbon atoms needed to make the dot=$0.5016 \times {10^{17}}$
On further solving, we get-
The total number of carbon atoms needed to make the dot=$5.016 \times {10^{16}} = 5 \times {10^{16}}$ (approx.)
On comparing this with the value given in the question, we get-
$ \Rightarrow 5 \times {10^x} = 5 \times {10^{16}}$
On solving, we get-
$ \Rightarrow {10^x} = {10^{16}}$
On further solving, we get-
$ \Rightarrow x = 16$

The value of x is $16$.

Note:Here the student may get the wrong value if he/she does not convert the unit of mass from microgram to gram. We convert the unit microgram into gram because the unit of atomic mass is also in gram so it is easier to solve the equation.