Problem 49 - Master Resource Book in JEE Main Physics

Question

One mole of an ideal gas at standard temperature and pressure occupies \(22.4 \mathrm{~L}\) (molar volume). The ratio of molar volume to the atomic volume of a mole of hydrogen? (Take the size of hydrogen molecule to be about \(1 \mathrm{~A}\) ) (a) \(9.1 \times 10^{4}\) (b) \(6 \times 10^{4}\) (c) \(7.1 \times 10^{4}\) (d) \(8.1 \times 10^{5}\)

Step-by-Step Solution

Answer

Option (c) \(7.1 \times 10^{4}\).

Step 1: Calculate Atomic Volume

First, calculate the atomic volume of one hydrogen molecule using its dimensions. The volume of a sphere (approximating a hydrogen molecule) is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the hydrogen atom. Given that the size of the hydrogen molecule is about \(1\ \text{Å}\) or \(0.1\ \text{nm}\), and considering the diameter is given, we need to use radius \( r = 0.5 \text{Å} = 0.05\ \text{nm} \). Thus, convert \( r \) into meters: \( r = 0.05 \times 10^{-9} \text{m} \).

Step 2: Determine Volume of a Hydrogen Molecule

Now calculate the volume of one hydrogen molecule:\[V = \frac{4}{3} \pi (0.05 \times 10^{-9})^3.\]Upon calculation, \( V \approx 5.24 \times 10^{-31} \text{m}^3 \).

Step 3: Calculate Atomic Volume of a Mole

To find the atomic volume of a mole of hydrogen, multiply the volume of one hydrogen molecule by Avogadro's number \(6.022 \times 10^{23}\ mol^{-1}\):\[V_{\text{mole}} = 5.24 \times 10^{-31} \times 6.022 \times 10^{23} \approx 3.15 \times 10^{-7} \text{m}^3.\]

Step 4: Calculate the Ratio

The molar volume of an ideal gas at STP is given as \(22.4\ \text{L} = 22.4 \times 10^{-3} \text{m}^3\). Compute the ratio by dividing the molar volume by the atomic volume:\[\text{Ratio} = \frac{22.4 \times 10^{-3}}{3.15 \times 10^{-7}} \approx 7.11 \times 10^{4}.\]

Step 5: Select the Closest Option

Compare the calculated ratio \(7.11 \times 10^{4}\) with the given options. The closest option is (c) \(7.1 \times 10^{4}\).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas. The equation is expressed as \( PV = nRT \), where \( P \) stands for pressure, \( V \) is volume, \( n \) represents the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This law provides a useful approximation for the behavior of real gases under many conditions, such as low pressure and high temperature.

One key application of the Ideal Gas Law is determining the molar volume of gases. Molar volume is the volume occupied by one mole of gas at a given temperature and pressure, typically standard temperature and pressure (STP). At STP, which is defined as 0 °C (273.15 K) and 1 atm pressure, the molar volume of an ideal gas is 22.4 liters.

This law is instrumental in calculations involving gases, as it allows scientists to predict changes in gas behavior when certain conditions are altered, like temperature or pressure. Understanding and applying the Ideal Gas Law is crucial in both theoretical studies and practical applications involving gases.

Atomic Volume

Atomic volume refers to the volume occupied by one mole of atoms or molecules in a substance. When dealing with gases, particularly at the atomic level, it's useful to consider the volume a single atom or molecule would occupy to understand the material's characteristics.

The atomic volume is calculated for a single molecule and then expanded to a mole by multiplying it by Avogadro's Number. For a hydrogen molecule, which is a diatomic molecule (H₂), the atomic volume calculation involves approximating the molecule as a sphere. Using its diameter, you calculate the volume with the formula for the volume of a sphere:

• Volume, \( V = \frac{4}{3} \pi r^3 \).

Given that the size of the hydrogen molecule is approximately 1 Å (angstrom), you find its radius and then compute its volume at the atomic level. This figure is critical for comparisons against larger volumes, such as the molar volume at STP.

Understanding atomic volume helps in grasping how individual molecules contribute to the overall volume of a gas, particularly in exercises that involve comparing this with standardized volumes like the molar volume.

Hydrogen Molecule

Hydrogen molecules, represented as H₂, are the simplest and smallest molecules, composed of two hydrogen atoms. They serve as an excellent example for many basic chemical calculations because of their simplicity.

In calculating properties related to the hydrogen molecule, one often approximates its dimensions, as the molecule is small (around 1 Å in diameter, equivalent to 0.1 nm). This approximation is useful in a variety of calculations, such as determining their volume and properties like the atomic volume mentioned earlier.

Hydrogen is the most abundant element in the universe, and its molecules exhibit key characteristics fundamental to physics and chemistry. For instance:

• They are involved in basic reactions like hydrogen bonding, crucial in many chemical and biological processes.
• They provide a foundational base for principles like Avogadro's hypothesis, aiding in understanding molecular interactions and behaviors.

Recognizing the characteristics and roles of the hydrogen molecule helps in tackling larger concepts such as gas behavior and atomic interactions in chemistry.

Avogadro's Number

Avogadro's Number is a key concept in chemistry, indicating the number of particles, which could be atoms, molecules, ions, etc., present in one mole of a substance. This number is approximately \(6.022 \times 10^{23}\).

Named after the scientist Amedeo Avogadro, this number makes it feasible to understand the mole as a bridge between an atom's scale and a macroscopic scale in chemistry. For example, when determining the atomic volume of a gas like hydrogen, you multiply the volume of a single molecule by Avogadro's Number to find the total volume one mole of such molecules would occupy.

This number is essential since it links atomic theory to measurable quantities in the laboratory, enabling scientists to make accurate predictions and calculations about chemical reactions and properties of substances. Understanding Avogadro's Number can make complex calculations involving large quantities of small particles more intuitive and accessible.