Problem 179 - Master Resource Book in JEE Main Physics

Question

Two balloons are filled, one with pure helium gas and the other by air, respectively. If the pressure and temperature of these balloons are same then the number of molecules per unit volume is [UP SEE 2006] (a) more in the helium filled balloon (b) same in both balloons (c) more in air filled balloon (d) in the ratio of \(1: 4\)

Step-by-Step Solution

Answer

(a) more in the helium filled balloon

Step 1: Understanding Ideal Gas Law

The ideal gas law relates the pressure \( P \), volume \( V \), and temperature \( T \) of a gas to the number of moles \( n \) through the equation \( PV = nRT \), where \( R \) is the ideal gas constant. When pressure and temperature are constant in two different gases, the ratio of volumes is the same as the ratio of the number of moles.

Step 2: Relating Moles to Molecules

The number of moles \( n \) is related to the number of molecules \( N \) by Avogadro's number \( N_A \). Thus, \( n = \frac{N}{N_A} \). If two balloons have the same pressure and temperature, the number of molecules per unit volume will be directly related to the density of the respective gases.

Step 3: Considering Gas Densities

Helium is a monoatomic gas with a lower molar mass compared to the diatomic molecules predominantly found in air (mainly nitrogen and oxygen). Under the same conditions, helium will have more molecules per unit volume because mole count is inversely proportional to the molar mass for the same pressure and temperature; lighter gases will have more molecules per unit volume.

Key Concepts

Helium and Air Comparison

When comparing a balloon filled with helium gas and another with air under the same pressure and temperature, it's crucial to consider the composition of both gases. Helium is a monoatomic gas, consisting of individual helium atoms. It's much lighter than the diatomic molecules found in air, primarily nitrogen (N₂) and oxygen (O₂). This difference in molar mass, where helium weighs around 4 g/mol compared to approximately 28 g/mol for nitrogen, is significant. Since both balloons are under the same conditions, the ideal gas law tells us the number of moles is inversely proportional to the molar mass, given a constant volume. Thus, with helium being lighter, the balloon filled with helium will contain more moles, and consequently more molecules per unit volume, than the air-filled balloon. Hence, the molecular density is greater in the helium-filled balloon.

Molecular Density

Molecular density refers to the number of molecules present in a given volume of gas. According to the ideal gas law, the density depends on the molar mass and the number of moles. Under identical conditions of pressure and temperature, gases with a lighter molar mass will always exhibit higher molecular density. This is because:

• Lighter gases occupy the same volume as heavier gases with more molecules.
• The mass of gas affects how many molecules fit into a given space.

To calculate molecular density, we divide the total number of molecules by the volume they occupy. As seen, helium's lower molar mass compared to air results in a higher molecular density under the same conditions.

Avogadro's Number

Avogadro's number is a fundamental constant used in chemistry that defines the number of constituent particles, usually atoms or molecules, per mole of a substance. This number is approximately 6.022 x 10²³. Avogadro's principle states that equal volumes of all gases, at the same temperature and pressure, have the same number of molecules. This principle is applied in comparing the molecular density of helium and air. It highlights that despite different molar masses, gases have the same number of molecules per unit volume if they have equal temperature and pressure. In practice, Avogadro's number allows us to convert between the macroscopic measurements of moles and the microscopic count of particles in a given sample. It's essential in calculating molecular densities, as knowing the moles of gas provides insight into the total number of molecules present.