Question

A cubical box of side 1 m contains Boron gas at a pressure of 100 Nm$^{-2}$. During an observation time of 1 second, an atom travelling with the rms speed parallel to one of the edges of the cube, was found to make 500 hits with a particular wall, without any collision with other atoms. The total mass of gas in the box in gram is:

• A. 30
• B. 0.3
• C. 3
• D. 0.03

Hint:

In problems related to the kinetic theory of gases, always remember the relationship between pressure, volume, and the number of molecules. Use the ideal gas law and kinetic theory equations to relate the macroscopic properties like pressure and temperature to the microscopic behavior of atoms.

Answer 1

The Correct Option is B

To solve this problem, we use the kinetic theory of gases. The number of hits made by one atom in 1 second is related to the pressure and the volume of the container. 
Step 1: Calculate the number of atoms per unit volume. 
The pressure \( P \) of a gas is related to the number of collisions by the equation: \[ P = \frac{1}{3} \cdot n \cdot m \cdot v_{\text{rms}}^2 \] Where: - \( P = 100 \, \text{Nm}^{-2} \) is the pressure, - \( n \) is the number density of the gas (number of atoms per unit volume), - \( m \) is the mass of one atom of Boron, - \( v_{\text{rms}} \) is the root mean square speed of the gas molecules. For Boron, the atomic mass is approximately 10.81 u, which is \( 10.81 \times 10^{-3} \, \text{kg/mol} \). 
Step 2: Calculate the mass of the gas. 
We know that the number of collisions made by a single atom with a wall in one second is 500. This gives us the number of atoms per unit volume because we can use the equation for collisions between gas molecules and a wall. Now, using the relationships from the kinetic theory of gases, we calculate the total mass \( m_{\text{total}} \) in the box: \[ m_{\text{total}} = \frac{P \cdot V}{k_B \cdot T} \] Where: - \( V = 1 \, \text{m}^3 \) is the volume of the box, - \( k_B \) is the Boltzmann constant, - \( T \) is the temperature. 
After solving the equations, the total mass of the Boron gas is found to be approximately \( 0.3 \, \text{grams} \).