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A silver ornament of mass ‘m’ gram is polished with gold equivalent to 1% of the mass of silver. Then the ratio of number of atoms of gold and silver in the ornament is
A. 108:197
B. 1:182.41
C. 108:19700
D. Both 2 and 3

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Last updated date: 19th Jun 2024
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Answer
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Hint: The number of atoms in any substance (element) can be determined by mole concept. One mole is defined as the amount of substance that contains as many entities (atoms, molecules, or other particles) as these are in exactly 12g of ${C^{12}}$ isotope.

Complete step by step answer:
Given, mass of silver = m g
Mass of gold $ = 1\% $ of mass of silver
$ = \left( {\dfrac{1}{{100}} \times m} \right)g$

$ = \dfrac{m}{{100}}g$
We know that, Number of atoms = Number of moles $\times$ Avogadro’s constant
By using the formula,
To find the Number of atoms of silver, we have
Number of atoms of silver = Number of moles of silver $\times$ Avogadro’s constant

 = $\dfrac{{Weight}}{{Mass}} \times Avogadro’s\,Constant$

By putting the value of weight of silver, molar mass, Avogadro’s constants
$\therefore$ Number of atoms of silver = $\left( {\dfrac{M}{{108}} \times 6.022 \times {{10}^{23}}} \right)$ atoms…………… (i) (Where molar mass of Ag = 108g)
To find the number of atoms of gold, we have
Number of gold atoms = Number of moles gold $\times$ Avogadro’s constant

=$\dfrac{{Weight}}{{Molar{\text{ mass}}}} \times\,Avogadro’s\,Constant$

=$\left( {\dfrac{M}{{100 \times 197}} \times 6.022\times {{10}^{23}}} \right)$ atoms of gold……………… (ii) (Where, molar mass of gold = 197g)
From equation (i) and (ii)
Dividing (ii) from (i), we will get,

$\dfrac{{N{\text{umber of atoms of gold}}}}{{Number{\text{ of atoms of silver}}}} = \dfrac{{\dfrac{M}{{100 \times 197}} \times 6.022 \times {{10}^{23}}}}{{\dfrac{M}{{108}} \times 6.022 \times {{10}^{23}}}}$

= ${\dfrac{M}{19700}} \times {\dfrac{108}{M}}$

= ${\dfrac{108}{19700}}$

$\therefore$ The ratio of number of atoms of gold to the number of atoms of silver is 108:19700

Hence, the correct option is (C).

Note: A system containing a specified number $\left( {6.022 \times {{10}^{23}}} \right)$ of elementary entities is said to contain 1 mole of entities.
Thus 1 mole of an iron sample means that the sample contains $6.022 \times {10^{23}}$ atoms of iron. All the calculations in chemistry that include the number of atoms are based on mole concept. To find out the mole of any particular substance we must know the atomic mass or molecular mass of the substance.