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# A signature written in carbon pencil weighs $1mg$ . What is the number of carbon atoms present in the signature?A.$5.02 \times {10^{23}}$B.$5.02 \times {10^{20}}$C.$6.02 \times {10^{20}}$D.$0.502 \times {10^{20}}$

Last updated date: 25th Jun 2024
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Hint:Here we must know the relation between molecular mass and Avogadro’s number. Avogadro’s number is actually a constant. The mass of one mole of any substance is equal to the molecular mass or molar mass expressed in grams. One mole of any substance contains Avogadro’s number of particles.

-The number of entities or particles that are present in one mole of any substance is equal to $6.023 \times {10^{23}}$ . This is known as the Avogadro’s number or Avogadro’s constant. It is basically used to calculate the number of particles in any given sample.
-One mole is the amount of any substance that contains as many particles that are present in exactly $12g$ of isotope of carbon $\left( {^{12}C} \right)$ .
-So we can say that one mole of hydrogen atoms contains $6.023 \times {10^{23}}$ atoms. Similarly one mole of water contains $6.023 \times {10^{23}}$ molecules.
-When we express the mass of one mole of any substance in grams then that mass is called a molar mass. So we can say that the molar mass of water is $18gmo{l^{ - 1}}$ .
Also $18g$of water contains $6.023 \times {10^{23}}$ atoms.
-Now the molar mass of carbon is $12gmo{l^{ - 1}}$ . So we can say that $12g$ of carbon contains Avogadro’s number of particles.
-So the number of carbon atoms in $1mg$ of carbon can be calculated as –
$1mg = 0.001g$
$n = \dfrac{{given\,mass \times Avogadro\,number}}{{Molar\,mass\,of{\,^{12}}C}}$
$n = \dfrac{{0.001 \times 6.023 \times {{10}^{23}}}}{{12}}$
Where $n$ is the number of carbon atoms.
So the answer of the above equation comes out to be $0.502 \times {10^{20}}$ .
So the correct answer is option D.

Note:
Avogadro’s number is a constant and its value does not change. Avogadro’s law states that equal volume of all gases under similar conditions of temperature and pressure contains equal numbers of molecules.