The Question is from neet-chemistry chemical-thermodynamics kinetic-theory, and it was created by the author Mr. S M HUSSAIN

Question

A mixture of two diatomic ideal gases A and B, having molecular masses $m_A$ and $m_B$ respectively, is in thermal equilibrium.  If $v_{rms,A}$ and $v_{rms,B}$ represent their root mean square speeds, what is the ratio of their most probable speeds ($v_{mp,A}/v_{mp,B}$)?

Mr. S M HUSSAIN · Accepted Answer

## CORE INFORMATION
✓ **Answer**: C
## CONCEPT FRAMEWORK
 **Primary Concept**: Kinetic Theory of Gases, specifically the relationship between molecular mass and most probable speed.
 **Related Topics**: Root mean square speed, average speed, Maxwell-Boltzmann distribution.
 **Prerequisites**: Understanding of ideal gas behavior, temperature, and molecular mass.
## SOLUTION PATHWAY
**Given Information**:
  Two diatomic ideal gases A and B with molecular masses $m_A$ and $m_B$ respectively.
  The gases are in thermal equilibrium (same temperature).
  $v_{rms,A}$ and $v_{rms,B}$ are their root mean square speeds.
**Method & Steps**:
 1. Recall the formula for the most probable speed ($v_{mp}$) of a gas molecule:
  $v_{mp} = \sqrt{\frac{2kT}{m}}$, where k is the Boltzmann constant, T is the temperature, and m is the molecular mass.
 2. For gas A, $v_{mp,A} = \sqrt{\frac{2kT}{m_A}}$ and for gas B, $v_{mp,B} = \sqrt{\frac{2kT}{m_B}}$.
 3. Find the ratio of their most probable speeds:
  $\frac{v_{mp,A}}{v_{mp,B}} = \frac{\sqrt{\frac{2kT}{m_A}}}{\sqrt{\frac{2kT}{m_B}}} = \sqrt{\frac{2kT}{m_A} 	imes \frac{m_B}{2kT}} = \sqrt{\frac{m_B}{m_A}}$
  Result: The ratio of their most probable speeds is $\sqrt{\frac{m_B}{m_A}}$.
 4. Verification: The formula shows that lighter molecules (smaller m) have higher most probable speeds.
 ## OPTION ANALYSIS
A :x: Incorrect. This is the ratio of molecular masses, not the ratio of most probable speeds.
B :x: Incorrect. This is related to the ratio of rms speeds, not most probable speeds.
 C ✓: Correct. This accurately reflects the relationship between most probable speed and molecular mass as derived above.
D :x: Incorrect.  Only if $m_A = m_B$ would the most probable speeds be equal.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Confusing most probable speed with root mean square speed or average speed: Remember the different formulas and their derivations.
  2.Incorrectly manipulating the square root in the ratio calculation: Pay close attention to the algebra when simplifying the expression.
 **Quick Tips**:
  1.Remember the formula for most probable speed directly: $v_{mp} = \sqrt{\frac{2kT}{m}}$.
  2.Relate the inverse square root relationship between most probable speed and molecular mass conceptually: Lighter molecules move faster.

Mr. S M HUSSAIN · Answer

$m_A/m_B$

Mr. S M HUSSAIN · Answer

$\sqrt{m_A/m_B}$

Mr. S M HUSSAIN · Answer

The correct Option is "$\sqrt{m_B/m_A}$"

Mr. S M HUSSAIN · Answer

$1$

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