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

(d) 72%

Step 1: Calculate the Volume of an Argon Atom

To calculate the volume of a single argon atom, we use the formula for the volume of a sphere, which is \(V = \frac{4}{3} \pi r^3\). The radius \(r\) of the argon atom is given as \(1.54 \times 10^{-8} \mathrm{~cm}\). So, the volume \(V\) is:\[ V = \frac{4}{3} \pi (1.54 \times 10^{-8})^3 \text{ cm}^3 \].

Step 2: Calculate the Volume of Argon Atoms in 1 mL of Solid Argon

Since 1 mL of solid argon has a mass of \(1.65 \mathrm{~g}\), and the atomic weight of argon is \(40\), the number of moles in 1 mL is \( \frac{1.65}{40} \). The number of atoms is the number of moles multiplied by Avogadro's number (\(6.022 \times 10^{23}\)). This gives:\[ \text{Number of argon atoms} = \frac{1.65}{40} \times 6.022 \times 10^{23} \].The total volume occupied by these atoms is:\[ \text{Total Volume} = \text{Number of argon atoms} \times V \text{ (from Step 1)} \].

Step 3: Calculate the Percentage of Empty Space

To find the percentage of empty space, we need to compare the total volume occupied by the argon atoms with the volume of the solid. The percentage of empty space is calculated by:\[ \text{Percentage of Empty Space} = \left(1 - \frac{\text{Total Volume of Atoms}}{1 \text{ mL}}\right) \times 100\% \].Substitute the values from Step 2 to find the percentage.

Key Concepts

Volume of a Sphere

To understand how much space an atom occupies, we first need to calculate its volume, often considered a sphere for simplicity. The formula for the volume of a sphere is:\[ V = \frac{4}{3} \pi r^3 \]where \(r\) is the radius of the sphere. For an argon atom, the radius is given as \(1.54 \times 10^{-8} \text{ cm}\). This is a very small figure, as atoms are incredibly tiny. Plugging this number into our formula, we get:\[ V = \frac{4}{3} \pi (1.54 \times 10^{-8})^3 \text{ cm}^3 \]By calculating the volume, you can understand the space that a single argon atom occupies. Understanding this concept is crucial as it forms part of the process to determine how much actual space argon atoms take in a solid structure. Often in solids, atoms are packed closely, but space between atoms may not be negligible, leading to empty spaces.

Atomic Structure

The atomic structure plays a significant role when considering density and empty space in a solid. Argon atoms, like many others, consist of a nucleus surrounded by an electron cloud. However, these electron clouds do not always perfectly fill in the space within a solid structure, leading to empty spaces—or voids—in the material. In a solid form, atoms pack together, but they do not always occupy all the available space, leaving gaps between them. This relationship reflects how tightly a solid's particles are packed, impacting the solid's observable density. Generally, higher the density, lower is the empty space. Understanding atomic structure helps us visualize why the measured density of a material might be less than expected, due to these unoccupied volumes in its atomic structure.

Percentage Calculation

Percentage calculation is an essential tool in determining the efficiency and effectiveness of space usage in solids. To find out how much of a solid's volume is actually occupied by its atoms, you'll frequently calculate percentages. In the context of argon, this involves working out the fraction of the total volume the atoms themselves occupy within 1 mL of the solid.The formula to calculate the percentage of empty space is:\[\text{Percentage of Empty Space} = \left(1 - \frac{\text{Total Volume of Atoms}}{\text{Total Volume of Solid}}\right) \times 100\% \]First, calculate the total volume occupied by the argon atoms from their calculated volume. Divide this by the considered volume of the solid (1 mL in this example) to find the portion of the solid's space taken by atoms. Subtract this value from 1, then multiply by 100 to convert the value into a percentage. This gives a clear insight into how much of the solid is just empty space.