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The International Avogadro Coordination project created the world's most perfect sphere using Silicon in its crystalline form. The diameter of the sphere is $9.4 \,cm$ with an uncertainty of $0.2 \,nm$. The atoms in the crystals are packed in cubes of side $a$. The side is measured with a relative error of $2 \times 10^{-9}$, and each cube has $8$ atoms in it. Then, the relative error in the mass of the sphere is closest to (assume molar mass of Silicon and Avogadro's number to be known precisely)
$6.4 \times 10^{-9}$
$4.0 \times 10^{-10}$
$1.2 \times 10^{-8}$
$5.0 \times 10^{-8}$
Solution
(C)
$\text { Mass } \Rightarrow \frac{\frac{4}{3} \pi\left(\frac{ d }{2}\right)^3}{ a ^3} \times 8 \times\left(\frac{\text { Molar Mass }}{N_{ A }}\right)$
Take $\log$ and differentiate we get
$\frac{\Delta m }{ m }=3 \frac{\Delta d }{ d }+3 \frac{\Delta a }{ a }$
$\frac{\Delta m }{ m }=3 \times\left(\frac{0.2 \times 10^{-9}}{9.4 \times 10^{-2}}\right)+3\left(2 \times 10^{-9}\right)$
$\frac{\Delta m }{ m }= 6.3 \times 10^{-9}+6 \times 10^{-9} \Rightarrow 12.3 \times 10^{-9}$
$\frac{\Delta m }{ m }=1.2 \times 10^{-8}$
