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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.$0.502 \times {10^{20}}$

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Last updated date: 25th Jun 2024
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Answer
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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.

Complete step by step answer:
-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.