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Question

The most probable speed of a gas molecule at a certain temperature is 400 m/s.  Which of the following is closest to its root mean square speed?

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
 **Primary Concept**: Kinetic theory of gases, specifically the relationship between most probable speed, average speed, and root mean square speed (RMS speed) of gas molecules.
 **Related Topics**: Maxwell-Boltzmann distribution, temperature and molecular kinetic energy.
 **Prerequisites**: Understanding of statistical mechanics and basic probability concepts.
## SOLUTION PATHWAY
**Given Information**:
  Most probable speed ($v_p$) = 400 m/s
**Method & Steps**:
 1. Relate most probable speed and root mean square speed.
  Reasoning: For an ideal gas, the most probable speed ($v_p$), average speed ($\bar{v}$), and root mean square speed ($v_{rms}$) are related by the following equations:
  $v_p = \sqrt{\frac{2kT}{m}}$
  $\bar{v} = \sqrt{\frac{8kT}{\pi m}}$
  $v_{rms} = \sqrt{\frac{3kT}{m}}$
  where k is Boltzmann's constant, T is the absolute temperature, and m is the mass of a gas molecule.
  Formula/Rule:  $v_{rms} = \sqrt{\frac{3}{2}} v_p$
 2. Calculate the root mean square speed.
  Working: Substituting the given most probable speed:
  $v_{rms} = \sqrt{\frac{3}{2}} 	imes 400 \, m/s \approx 489.9 \, m/s$
  Key Point: The ratio between $v_{rms}$ and $v_p$ is a constant ($\sqrt{\frac{3}{2}} \approx 1.225$).
 3. Compare the calculated RMS speed with the given options.
  Result: The calculated RMS speed is approximately 490 m/s.
  Verification: The approximation is reasonable given the context of the question.
 ## OPTION ANALYSIS
A :x: 400 m/s is the most probable speed, not the RMS speed.
B ✓: 490 m/s is the closest to the calculated RMS speed of approximately 490 m/s.
C :x: 320 m/s is significantly lower than the expected RMS speed.
D :x: 560 m/s is too high compared to the calculated RMS speed.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Confusing most probable speed, average speed, and RMS speed: Remember the relationships between these three speeds and their corresponding formulas.
  2.Incorrectly using the relationships between the speeds: Ensure that the correct ratios are used for the calculations.
 **Quick Tips**:
  1.Remember the approximate ratio between $v_{rms}$ and $v_p$ ($\sqrt{\frac{3}{2}} \approx 1.225$) to quickly estimate the RMS speed.
  2.Visualize the Maxwell-Boltzmann distribution to understand the relative magnitudes of $v_p$, $\bar{v}$, and $v_{rms}$.

P.NARASIMHA · Answer

400 m/s

P.NARASIMHA · Answer

The correct Option is "490 m/s"

P.NARASIMHA · Answer

320 m/s

P.NARASIMHA · Answer

560 m/s

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