Kinetic Energy and Molecular Velocities - Free MCQ Practice Test with solutions,

The ratio of root mean square speed, represented as u_rms to the most probable speed, represented as u_mp is always the same for identical conditions and same gas. It is
 

1. Root mean square speed equation for Argon:
   The formula for the root mean square speed of Argon is:
   vₓₘₛ,ₐᵣ = √(3kT / mₐᵣ),
   where k is the Boltzmann constant, T is the temperature in Kelvin, and mₐᵣ is the mass of Argon.

2. Average speed equation for Helium:
   The formula for the average speed of Helium is:
   vₐᵥₓ,ₕₑ = √(8kTₕₑ / (π * mₕₑ)),
   where Tₕₑ is the temperature of Helium in Kelvin, and mₕₑ is the mass of Helium.

3. Equating the two speeds:
   Since the root mean square speed of Argon equals the average speed of Helium, the two equations are set equal, and solving for T gives:
   T = (8 * mₐᵣ * Tₕₑ) / (3 * π * mₕₑ).

4. Substituting values:

   • mₐᵣ = 39.9 u (atomic mass of Argon)

   • mₕₑ = 4.0 u (atomic mass of Helium)

   • Tₕₑ = 253.15 K (temperature of Helium)

After substituting these values, the temperature T for Argon is approximately 2143.43 K

 c) 1935.43 m/s

Corrected Solution: The root mean square speed v_rms = sqrt(3×R×T/M). Here R = 8.314 J·mol⁻¹·K⁻¹, T = 27 + 273 = 300 K, M = 2 g/mol = 0.002 kg/mol. Calculate 3×8.314×300 = 7482.6, divide by 0.002 gives 3,741,300, then sqrt(3,741,300) ≈ 1933.9 m/s, which rounds to 1935.43 m/s.

The root means square speed of particles is nothing but the square root over the sum of squares of the particle’s speeds by a total number of particles. So by substituting, √3² + 4² + 5²/3 = 4.082 m/s.

According to the formula, the root mean square speed is greater than the average speed and the average speed is greater than the most probable speed at given identical conditions and for the same gas.

The formula of average speed is given bywhere R is universal gas constant, T is a temperature in Kelvin and M is the mass in kilograms. From the formula, we understand that the average speed is inversely proportional to the root over the mass. As hydrogen has the least mass among the options it has the highest average speed.

The ratio of root mean square speed to the mean probable speed is 1.224. So the above statement is considered to be wrong. The ratio between the main probable speed and the average speed and root mean square speed is 1 : 1.128 : 1.224.