Chapter 9: 43E (page 405)

Using the Maxwell speed distribution, determine the most probable speed of a particle of mass min a gas at temperature T
How does this compare with v_rms ? Explain.

Short Answer

Expert verified

The most probable speed is $\sqrt{\frac{2 k_{B} T}{m}}$ .
The most probable speed is $\sqrt{\frac{2}{3}}$ times $v_{rms}$ .

Step by step solution

Maxwell Probability Distribution.

$P (v) = {(\frac{m}{2 {πk}_{B} T})}^{\frac{3}{2}} 4 {πv}^{2} e^{- \frac{{mv}^{2}}{2 k_{B} T}}$ …..(1)

Where,

m is the mass of the particle.

vis velocity of particle.

T is temperature.

k_B is Boltzmann constant.

$\begin{array}{rcl} \frac{d P}{d v} & = & \frac{d}{dv} [\sqrt{\frac{2}{π}} {(\frac{m}{k_{B} T})}^{\frac{3}{2}} v^{2} e^{- \frac{{mv}^{2}}{2 k_{B} T}}] \\ \frac{d P}{d v} & = & \sqrt{\frac{2}{π}} {(\frac{m}{k_{B} T})}^{\frac{3}{2}} \frac{d}{dv} [v^{2} e^{- \frac{{mv}^{2}}{2 k_{B} T}}] \\ \frac{d P}{d v} & = & \sqrt{\frac{2}{π}} {(\frac{m}{k_{B} T})}^{\frac{3}{2}} [2 {ve}^{- \frac{{mv}^{2}}{2 k_{B} T}} - (2 v) (\frac{m}{k_{B} T}) v^{2} e^{- \frac{{mv}^{2}}{2 k_{B} T}}] \\ \frac{d P}{d v} & = & 2 \sqrt{\frac{2}{π}} {(\frac{m}{k_{B} T})}^{\frac{3}{2}} [1 - \frac{{mv}^{2}}{2 k_{B} T}] {ve}^{- \frac{{mv}^{2}}{2 k_{B} T}} \\ 0 & = & 2 \sqrt{\frac{2}{π}} {(\frac{m}{k_{B} T})}^{\frac{3}{2}} [1 - \frac{{mv}^{2}}{2 k_{B} T}] {ve}^{- \frac{{mv}^{2}}{2 k_{B} T}} \\ 0 & = & 1 - \frac{{mv}^{2}}{2 k_{B} T} \\ \frac{{mv}^{2}}{2 k_{B} T} & = & 1 \\ v^{2} & = & \frac{2 k_{B} T}{m} \\ v & = & \sqrt{\frac{2 k_{B} T}{m}} \end{array}$

Therefore, the most probable speed is $\sqrt{\frac{2 k_{B} T}{m}}$ .

Mathematical Expression of rms Speed.

$v_{rms} = \sqrt{\frac{3 k_{B} T}{m}}$

Rearrange for $\sqrt{\frac{k_{B} T}{m}}$ ,

$\sqrt{\frac{k_{B} T}{m}} = \frac{v_{rms}}{\sqrt{3}}$

The mathematical expression for the most probable speed is,

$v = \sqrt{\frac{2 k_{B} T}{m}}$

Substitute $\frac{v_{rms}}{\sqrt{3}}$ for $\sqrt{\frac{k_{B} T}{m}}$ ,

$\begin{array}{rcl} v & = & \sqrt{2} \frac{v_{rms}}{\sqrt{3}} \\ = & \sqrt{\frac{2}{3}} v_{rms} \end{array}$

Therefore, the most probable speed is $\sqrt{\frac{2}{3}}$ times v_rms.

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Using the Maxwell speed distribution, determine the most probable speed of a particle of mass min a gas at temperature T
How does this compare with v_rms ? Explain.

Short Answer

Step by step solution

Maxwell Probability Distribution.

Mathematical Expression of rms Speed.

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Using the Maxwell speed distribution, determine the most probable speed of a particle of mass min a gas at temperature THow does this compare with vrms ? Explain.

Short Answer

Step by step solution

Maxwell Probability Distribution.

Mathematical Expression of rms Speed.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Measurements

Oscillations

Physics of Motion

Rotational Dynamics

Wave Optics

Study anywhere. Anytime. Across all devices.

Using the Maxwell speed distribution, determine the most probable speed of a particle of mass min a gas at temperature T
How does this compare with v_rms ? Explain.