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Question

Consider a gas of non-interacting molecules with mass $m$ at a temperature $T$. If the pressure is $P$ and the volume is $V$, what is the average speed of the molecules?

A.RAMA NAGESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: C
## CONCEPT FRAMEWORK
 **Primary Concept**: Kinetic Theory of Gases
 **Related Topics**: Root Mean Square Speed, Average Speed, Maxwell-Boltzmann Distribution
 **Prerequisites**: Understanding of ideal gas law, kinetic energy, and statistical mechanics.
## SOLUTION PATHWAY
**Given Information**:
  Mass of molecule = $m$
  Temperature = $T$
  Pressure = $P$
  Volume = $V$
**Method & Steps**:
 1. Relate average kinetic energy to temperature.
  Reasoning: The average kinetic energy of a molecule in an ideal gas is directly proportional to the absolute temperature.
  Formula/Rule: $\frac{1}{2} m \langle v^2 angle = \frac{3}{2} k_B T$, where $\langle v^2 angle$ is the mean square speed and $k_B$ is the Boltzmann constant.
 2. Find the root mean square (rms) speed.
  Working: From the above equation, we get $\langle v^2 angle = \frac{3k_B T}{m}$. The rms speed is $v_{rms} = \sqrt{\langle v^2 angle} = \sqrt{\frac{3k_B T}{m}}$.
  Key Point:  The rms speed is not the same as the average speed.  The average speed requires integrating over the Maxwell-Boltzmann distribution.
 3. Determine the average speed.
  Result: The average speed, $v_{avg}$, for a gas of molecules is given by $v_{avg} = \sqrt{\frac{8k_B T}{\pi m}}$. Since $R = N_A k_B$ (where $R$ is the ideal gas constant and $N_A$ is Avogadro's number), and $M = N_A m$ (where $M$ is the molar mass), we can rewrite this as $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$.
  Verification: The units work out correctly to be m/s. The formula is consistent with the kinetic theory of gases.
 ## OPTION ANALYSIS
A :x: This is the rms speed, not the average speed.
B :x: This is related to the speed of sound in an ideal gas, but not the average molecular speed.
C ✓: This is the correct formula for the average speed of molecules in an ideal gas.
D :x: This expression relates pressure and density, but doesn't directly give the average speed.  It's related to the rms speed through the ideal gas law but not directly the average speed.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Confusing average speed with rms speed: Remember that the average speed is slightly lower than the rms speed.
  2.Incorrectly using the ideal gas law: The ideal gas law ($PV = nRT$) helps relate pressure, volume, temperature, and the number of moles, but it doesn't directly give the average speed.
 **Quick Tips**:
  1.Remember the key formula: $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$
  2.Understand the difference between average speed, rms speed, and most probable speed.  They are all related but distinct properties of the gas molecules.

A.RAMA NAGESWARARAO · Answer

$\sqrt{3RT/M}$

A.RAMA NAGESWARARAO · Answer

$\sqrt{2RT/M}$

A.RAMA NAGESWARARAO · Answer

The correct Option is "$\sqrt{8RT/\pi M}$"

A.RAMA NAGESWARARAO · Answer

$\sqrt{3P/\rho}$

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