Question

At what temperature will the RMS velocity of sulfur dioxide molecules at 400 K be the same as the most probable velocity of oxygen molecules?

• A. 600 K
• B. 200 K
• C. 400 K
• D. 300 K

Hint:

To compare velocities, use the formulas for RMS velocity and most probable velocity. The temperature ratio depends on the ratio of the molar masses of the two gases.

Answer 1

The Correct Option is D

Finding the Temperature at Which the RMS Velocity of SO₂ Matches the Most Probable Velocity of O₂

Step 1: Formula for RMS Velocity of a Gas

The root mean square (RMS) velocity is given by the formula:

v_rms = √(3kT/m)

where:

k is the Boltzmann constant

T is the temperature in Kelvin

m is the molar mass of the gas

For sulfur dioxide (SO₂), at 400 K, the RMS velocity is given by:

v_rms,SO₂ = √(3kT_SO₂/m_SO₂)

Step 2: Formula for Most Probable Velocity of a Gas

The most probable velocity (v_mp) is given by:

v_mp = √(2kT/m)

For oxygen (O₂), the most probable velocity is given by:

v_mp,O₂ = √(2kT_O₂/m_O₂)

Step 3: Setting the RMS Velocity of SO₂ Equal to the Most Probable Velocity of O₂

We equate the two velocities:

√(3kT_SO₂/m_SO₂) = √(2kT_O₂/m_O₂)

Step 4: Simplifying the Equation

Canceling out k and squaring both sides:

(3T_SO₂/m_SO₂) = (2T_O₂/m_O₂)

(T_SO₂/T_O₂) = (2m_SO₂)/(3m_O₂)

Step 5: Using Molecular Masses

The molar masses are:

SO₂: 64 g/mol

O₂: 32 g/mol

Substituting:

(T_SO₂/T_O₂) = (2 × 64)/(3 × 32) = 128/96 = 4/3

Step 6: Solving for T_O₂

Given T_SO₂ = 400 K:

400/T_O₂ = 4/3

T_O₂ = (3 × 400) / 4 = 300 K

Final Answer:

The temperature at which the RMS velocity of SO₂ at 400 K matches the most probable velocity of O₂ is:

300 K