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Question: The density of iron is \(7.87\,g\,c{m^{ - 3}}\). If the atoms are spherical and closely packed. The ...

The density of iron is 7.87 g cm−37.87\,g\,c{m^{ - 3}}. If the atoms are spherical and closely packed. The mass of an iron atom is 9.27×10−26 Kg9.27 \times {10^{ - 26}}\,Kg. What is the volume of an iron atom??
(i) 1.18×10−29 m31.18 \times {10^{ - 29}}\,{m^3}
(ii) 2.63×10−29 m32.63 \times {10^{ - 29}}\,{m^3}
(iii) 1.73×10−29 m31.73 \times {10^{ - 29}}\,{m^3}
(iv) 0.53×10−29 m30.53 \times {10^{ - 29}}\,{m^3}

Explanation

Solution

Density is the degree of compactness of a substance and is given by mass per unit volume. The density of iron and the mass of the iron atom is given. Manipulate the formula relating them all a bit, put the values and convert the values to suitable units to get the volume of the iron atom.

Complete step-by-step answer: The density of a substance is its mass per unit volume. The symbol most often used to represent density is ρ\rho . Mathematically, density of a substance is defined as mass divided by volume as: ρ=mV\rho = \dfrac{m}{V} , where ρ\rho is the density of the substance, mm is the mass of the substance, and VV is the volume of the same substance.
Therefore the equation can also be written as: V=mρ......(1)V = \dfrac{m}{\rho }......\left( 1 \right).
In the question, it is given that the density of iron is 7.87 g cm−37.87\,g\,c{m^{ - 3}}and the mass of iron atom is 9.27×10−26 Kg9.27 \times {10^{ - 26}}\,Kg.
But the values are not in the same unit and we need to convert them all either to SI units or CGS units. Since in the given options metremetre is given therefore we convert them all to SI units.
Therefore, density of iron = 7.87×10−3 Kg ×106m−3 = 7.87×103 Kg m−3 = \,7.87 \times {10^{ - 3}}\,Kg\, \times {10^6}{m^{ - 3}}\, = \,7.87 \times {10^3}\,Kg\,{m^{ - 3}}in SI units.
Mass of iron is already in SI units.
Therefore putting these values in equation (1)\left( 1 \right) we get, V=9.27×10−26 Kg7.87×103 Kg m−3 = 1.18×10−29 m3V = \dfrac{{9.27 \times {{10}^{ - 26}}\,Kg}}{{7.87 \times {{10}^3}\,Kg\,{m^{ - 3}}}}\, = \,1.18 \times {10^{ - 29}}\,{m^3}.

Hence the correct answer is (i) 1.18×10−29 m31.18 \times {10^{ - 29}}\,{m^3}.

Note: The most important part of this question is unit conversions. By looking at the question you might feel the question is easy and therefore in a hurry you can mix up SI and CGS units and do error in calculations. In order to avoid this proceed in a stepwise manner and convert to one unit before doing the calculations.