Problem 114 - A Complete Resource Book in Chemistry for JEE Main

Question

The density of solid argon is \(1.65 \mathrm{~g} / \mathrm{mL}\) at \(-233^{\circ} \mathrm{C}\). If the argon atom is assumed to be sphere of radius \(1.54 \times\) \(10^{-8} \mathrm{~cm}\), what percentage of solid argon is apprarently empty space? (Atomic wt of \(\mathrm{Ar}=40\) ) (a) \(32 \%\) (b) \(52 \%\) (c) \(62 \%\) (d) \(72 \%\)

Step-by-Step Solution

Answer

The percentage of solid argon that is empty space is 62%, so the answer is (c) 62%.

Step 1: Calculate the Volume of One Argon Atom

To find the volume of one argon atom, we use the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \). The radius \( r \) is given as \( 1.54 \times 10^{-8} \) cm. Thus, the volume \( V \) is \(V = \frac{4}{3} \pi (1.54 \times 10^{-8})^3 \approx 1.53 \times 10^{-23} \text{ cm}^3.\)

Step 2: Calculate the Mass of One Argon Atom

The atomic weight of argon is given as 40 g/mol. We use Avogadro's number, \(6.022 \times 10^{23}\) atoms/mol, to find the mass of one argon atom:\[\text{Mass of one atom} = \frac{40 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}} \approx 6.64 \times 10^{-23} \, \text{g/atom.}\]

Step 3: Calculate the Density of Argon Atoms

We divide the mass by the volume to find the density of a single argon atom:\[\text{Density} = \frac{6.64 \times 10^{-23} \, \text{g}}{1.53 \times 10^{-23} \, \text{cm}^3} \approx 4.34 \, \text{g/cm}^3.\]

Step 4: Calculate Percentage of Empty Space

The actual density of solid argon is given as 1.65 g/mL. The percentage of empty space is calculated by comparing the density of a single atom to the bulk density:\[\text{Empty Space Percentage} = \left(1 - \frac{1.65}{4.34}\right) \times 100\% \approx 62\%.\]

Key Concepts

Volume Calculation of Argon Atom

When measuring the volume of individual atoms like argon, we assume each atom to be a sphere. The formula for a sphere's volume is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) stands for the radius.
Given the radius of an argon atom as \( 1.54 \times 10^{-8} \) cm, we plug in this value:
\[ V = \frac{4}{3} \pi (1.54 \times 10^{-8})^3 \]
After calculating, the volume comes out to be approximately \( 1.53 \times 10^{-23} \text{ cm}^3 \).

Why is this calculation important? Knowing the volume of a single atom helps us understand how closely packed they would be in a solid state. In solid argon, these atomic spheres are tightly packed, but not without some space in between. Understanding the atom's volume is a crucial first step in understanding the entire structure of solid argon.

Atomic Weight of Argon

The atomic weight of an element describes its weight in grams per mole. For argon, the atomic weight is 40 g/mol. This value allows us to find the mass of a single argon atom.
To calculate this, we use Avogadro's number, \( 6.022 \times 10^{23} \) atoms/mol, which is a constant that tells us how many atoms are in a mole of a substance.
So, the mass of a single argon atom is:
\[ \text{Mass of one atom} = \frac{40 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}} \approx 6.64 \times 10^{-23} \, \text{g/atom} \]

This calculation is vital in determining how much mass an atom contributes to its solid form. Since we know the atomic weight and can calculate the mass of each atom, we are empowered to determine the density and evaluate empty spaces in solid structures.

Empty Space in Solids

Understanding how much of a solid's volume is occupied by atoms helps in knowing the material's density. However, not all of a solid's volume is filled with atoms.

In the case of solid argon, we know its density is 1.65 g/mL. After calculating, we find the density of individual argon atoms is 4.34 g/cm\(^3\).

By comparing the individual atom's density with the bulk density, we can estimate the empty space percentage:
\[ \text{Empty Space Percentage} = \left(1 - \frac{1.65}{4.34}\right) \times 100\% \approx 62\% \]

This result tells us that about 62% of solid argon is empty space. This calculation is essential to understand the packing and the nature of the solid material. Most materials are not as dense as they seem because of these gaps between the atoms.