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The Maxwell Distribution Laws - Practice Questions & MCQ

Edited By admin | Updated on Sep 25, 2023 25:23 PM | #NEET

Quick Facts

  • Various types of speeds of ideal gases is considered one the most difficult concept.

  • 92 Questions around this concept.

Solve by difficulty

At room temperature, a diatomic gas is found to have an r.m.s.  speed of 1930 ms-1. The gas is :

 

Which of the following statements is true for gas?

(i) For a certain temperature, the average speed is always greater than the most probable speed.

(ii) Ratio of Vrms: Vav: Vmp is: 1.77: 1.6:  1.41

Maxwell distribution curve at a particular temperature shows that

Concepts Covered - 2

Various types of speeds of ideal gases
  • Root mean square speed- It is defined as the square root of the mean of squares of the speed of different molecules.

             \text {ie. } v_{m s}=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+v_{4}^{2}+\ldots}{N}}=\sqrt{\bar{v}^{2}}

           1. As the Pressure due to an ideal gas is given as

                  P= \frac{1}{3}\rho \: v_{rms}^{2}

               \Rightarrow v_{m s}=\sqrt{\frac{3 P}{\rho}}=\sqrt{\frac{3 P V}{\text { Mass of gas }}}=\sqrt{\frac{3 R T}{M}}=\sqrt{\frac{3 k T}{m}}

                   Where 

R = Universal gas constant

M = molar mass

P = pressure due to gas

\rho = density

2. v_{m s} \ \alpha \ \ \sqrt{ T}  I.e With the rise in temperature, rms speed of gas molecules increases.

3. v_{m s} \ \alpha \ \ \frac{1}{ \sqrt{ M} } I.e With the increase in molecular weight, rms speed of the gas molecule decreases.

4. The  rms speed of gas molecules does not depend on the pressure of the gas (if the temperature remains constant)

  • Most probable speed-This is defined as the speed which is possessed by maximum the fraction of the total number of molecules of the gas.

          I.e   v_{m ps}=\sqrt{\frac{2 P}{\rho}}=\sqrt{\frac{2 R T}{M}}=\sqrt{\frac{2 k T}{m}}            

  •    Average speed-It is the arithmetic mean of the speeds of molecules in a gas at a given temperature.

                v_{a vg}=\frac{v_{1}+v_{2}+v_{3}+v_{4}+\ldots .}{N}

          and according to the kinetic theory of gases

                v_{a vg}=\sqrt{\frac{8 P}{\pi \rho}}=\sqrt{\frac{8}{\pi} \frac{R T}{M}}=\sqrt{\frac{8}{\pi} \frac{k T}{m}}

  • The relation between RMS speed, average speed, and most probable speed

                       V_{rms}> V_{avg}> V_{mps}

 

Maxwell's law

Maxwell’s Law - 

The  v_{rms} (Root mean square velocity) gives us a general idea of molecular speeds in a gas at a given temperature. So, it doesn't mean that the speed of each molecule is v_{rms}.

Many of the molecules have speed more than v_{rms} and many have speeds greater than v_{rms}. So, Maxwell derived an equation which describes the distribution of molecules in different speed as - 

                                                          \mathbf{d N=4 \pi \mathrm{N}\left(\frac{m}{2 \pi k T}\right)^{3 / 2} v^{2} e^{-\frac{m v^{2}}{2 k T}} d v}

                          \text { where, } d N=\text { Number of molecules with speeds between } v \text { and } v+dv

So, from this formula you have to remember few key points - 

                                                                            1.       \frac{dN }{dv}\propto N

                                                                           2.        \frac{dN}{dv} \propto v^2

                                 

Conclusions from this graph - 

1. This graph is between number of molecules at a particular speed and speed of these molecules.

2. You can observe that the  \frac{dN}{dv}  is maximum at most probable speed.

3.  This graph also represent that v_{rms}> v_{av}> v_{mp}.

4. This curve is asymmetric curve.

5. From this curve we can calculate number of molecules corresponds to that velocity range by calculating area bonded by this curve with speed axis. 

 

Effect of temperature on velocity distribution :

With rise of temperature the curve starts shifting right side and become broader as shown as -                    

                                                

Study it with Videos

Various types of speeds of ideal gases
Maxwell's law

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Books

Reference Books

Various types of speeds of ideal gases

Physics Part II Textbook for Class XI

Page No. : 325

Line : 49

Maxwell's law

Physics Part II Textbook for Class XI

Page No. : 331

Line : 14

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