The Maxwell Distribution Laws

Edited By Vishal kumar | Updated on Jul 02, 2025 05:33 PM IST

Maxwell’s distribution laws describe particle speed or velocity frequency distribution in gas at any given temperature. The collection of rules was produced by James Clerk Maxwell in the 19th century and provides a consistent means through which one can predict what gas molecules do. They are proof although that the majority is within some middle ground speed-wise for particles but always a few either at high velocities or lower velocities which move too slow hence making them important in describing different gases` physical characteristics.

This Story also Contains

What is Root mean square speed?
What is the Most probable speed?
What is Average speed?
What is Maxwell’s Law?
Solved Examples Based on The Maxwell Distribution Laws
Summary

The Maxwell Distribution Laws

In this article, we will cover the concept of the Maxwell distribution Laws. This topic we study in the kinetic theory of gases, which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), Twenty-six questions have been asked on this concept. And for NEET three questions was asked from this concept.

Let's read this entire article to gain an in-depth understanding of the Maxwell Distribution law, we have to discuss some basic terms related to it.

What is Root mean square speed?

It is defined as the square root of the mean of squares of the speed of different molecules.

ie. vms=v12+v22+v32+v42+…N=v¯2

As the Pressure due to an ideal gas is given as

P=13ρvrms2⇒vrms=3Pρ=3PV Mass of gas =3RTM=3kTm

Where,

R = Universal gas constant

M = molar mass

P = pressure due to gas

$\rho$ = density

vrmsαT I.e With the rise in temperature, rms speed of gas molecules increases.
vrmsα1M I.e With the increase in molecular weight, rms speed of the gas molecule decreases.
The rms speed of gas molecules does not depend on the pressure of the gas (if the temperature remains constant)

What is the 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;vmps=2Pρ=2RTM=2kTm

What is Average speed?

It is the arithmetic mean of the speeds of molecules in a gas at a given temperature.

vavg=v1+v2+v3+v4+…N

And according to the kinetic theory of gases

vavg=8Pπρ=8πRTM=8πkTm

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

Vrms>Vavg >Vmps

What is Maxwell’s Law?

The vrms (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 vrms.

Many of the molecules have speeds less than vrms and many have speeds greater than vrms. So, Maxwell derived an equation that describes the distribution of molecules at different speeds as -

dN=4πN(m2πkT)3/2v2e−mv22kTdv

where, dN= Number of molecules with speeds between v and v+dv

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

1. dNdv∝N
2. dNdv∝v2

Conclusions From This Graph

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

2. You can observe that the dNdv is maximum at the most probable speed.

3. This graph also represent that vrms>vav>vmp.

4. This curve is the asymmetric curve.

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

Effect of temperature on velocity distribution

With the rising of temperature, the curve starts shifting right side and becomes broader as shown as -

Solved Examples Based on The Maxwell Distribution Laws

Example 1: The value closest to the thermal velocity of a Helium atom at room temperature (300 K)ms−1 is : [kB=1.4×10−23 J/K;mHe=7× 10−27 kg]

1) 13000

2) 1300

3) 130000

4) 130

Solution:

v=3kTm=3∗1.4∗10−23∗3007∗10−27v=1.8∗106=1.3∗103 m/s

Hence, the answer is option (2).

Example 2: At room temperature, a diatomic gas is found to have an r.m.s. speed of 1930 ms^-1. The gas is :

1) H2
2) Cl2
3) O2
4) F2

Solution:

Root mean square velocity

Vrms=3RTM=3Pρ
At room temperature T = 300 K

For diatomic gas,

Vrms=3RTM=⇒M=3RTV2M=3×8.3×300(1930)2=2 g/mole

The diatomic gas is H2

Hence, the answer is the option (1).

Example 3: An ideal gas is enclosed in a cylinder at pressure of 2 atm and temperature 300 K. The mean time between two successive collisions is 6∗10−8 s. If the pressure is doubled and the temperature is increased to 500 K, the mean time (in sec) between two successive collisions will be close to:

1) 5×10−8 s
2) 2×10−8 s
3) 3×10−8 s
4) 4×10−8 s

Solution:

Root mean square velocity

Vrms=3RTM=3Pρ
wherein

R = Universal gas constant

M = molar mass

P = pressure due to gas

$\rho$ = density

Vrms∝TVrms∝ mean freepath timebetweensuccessivecollision
and mean free path =
Y=kT2πσ2P

Vrms∝YbVrms∝Tp×t…….1 but Vrms∝T……….2

From 1 and 2,

T∝TP×tt∝Tpt2t1=(T2T1)×(P1P2)=500300×P12P1=56t2=56t1t2≈4×10−8 s

Hence, the answer is option (4).

Example 4: A 15 g mass of nitrogen gas is enclosed in a vessel at a temperature 27∘C. The amount of heat (in kJ ) transferred to the gas, so that rms velocity of molecules is doubled is about: [ Take R = 8.3J/K mole ]

1) 10

2) 6

3) 14

4) 0.9

Solution:

Root mean square velocity

Vrms=3RTM=3Pρ

As gas is a closed vessel

Q=nCVΔTV∝T

RMS velocity is doubled

⇒>T2=4T1∴Q=nCV(4T1−T1)=nCV⋅3T1=1528×52R×3×300=10000 J=10 kJ

Hence, the answer is the option (1).

Example 5: A mixture of 2 moles of helium gas (atomic mass =4u ) and 1 mole of argon gas (atomic mass =40u ) is kept at 300 K in a container. The ratio of their ds [Vrms( helium )Vrms( argon )] is close to:

1) 3.16

2) 0.32

3) 0.45

4) 2.24

Solution:

Root mean square velocity -

Vrms=3RTMVrms(He)Vrms(Ar)=MArMHe=404=10

= 3.16

Hence, the answer is the option (1).

Summary

The Maxwell Distribution Laws provide a general description of the spread of gas-specific particle velocities and their effects on the pressure of a gas. This paper reports the existence of various molecular speeds with a greater number of particles that move at a mean speed while others have slow or very high velocities. This realization is important in understanding gas properties under various conditions such as pressure and heat.

Frequently Asked Questions (FAQs)

1. How is the average speed related to the most probable speed?

The average speed is slightly higher than the most probable speed in the Maxwell Distribution. Specifically, the average speed is about 1.128 times the most probable speed. This difference arises from the asymmetric shape of the distribution curve.

2. How does temperature affect the Maxwell Distribution?

As temperature increases, the Maxwell Distribution curve becomes wider and flatter, with its peak shifting to higher speeds. This means that at higher temperatures, molecules have a broader range of speeds and a higher average speed.

3. How does the Maxwell Distribution relate to the concept of thermal energy?

The Maxwell Distribution is directly related to thermal energy. The area under the distribution curve represents the total kinetic energy of the gas molecules, which is proportional to the temperature. Higher temperatures result in distributions with more high-speed molecules, indicating higher thermal energy.

4. How does the Maxwell Distribution explain gas diffusion?

The Maxwell Distribution helps explain gas diffusion by showing that molecules have a range of speeds. Faster molecules can travel further in a given time, leading to the spread of gas particles. The distribution of speeds contributes to the overall rate and pattern of diffusion.

5. Why is the Maxwell Distribution important in kinetic theory?

The Maxwell Distribution is crucial in kinetic theory because it provides a statistical description of molecular motion in gases. It helps explain macroscopic properties of gases, such as pressure and temperature, in terms of the microscopic behavior of individual molecules.

6. What is the root-mean-square (RMS) speed in the Maxwell Distribution?

The root-mean-square (RMS) speed is the square root of the average of the squared speeds of all molecules. It's slightly higher than both the average speed and the most probable speed, and is directly related to the temperature of the gas.

7. What is the connection between the Maxwell Distribution and the speed of sound in gases?

The speed of sound in a gas is related to the root-mean-square speed of its molecules, which is derived from the Maxwell Distribution. This connection helps explain why the speed of sound varies with temperature and the type of gas.

8. What role does the Maxwell Distribution play in understanding gas viscosity?

The Maxwell Distribution helps explain gas viscosity by describing the speed distribution of molecules. Viscosity arises from the transfer of momentum between layers of gas moving at different speeds, which is directly related to the molecular speed distribution.

9. How does the Maxwell Distribution contribute to our understanding of gas phase reactions in stars?

In stellar interiors, where temperatures are extremely high, the Maxwell Distribution helps predict the likelihood of nuclear fusion reactions. The high-energy tail of the distribution is particularly important, as it represents the particles with enough energy to overcome the Coulomb barrier and fuse.

10. Can the Maxwell Distribution explain why helium is used in airships instead of hydrogen, despite hydrogen being lighter?

Yes, the Maxwell Distribution shows that hydrogen molecules, being lighter, have a higher proportion of fast-moving molecules compared to helium. This means hydrogen is more likely to escape through the airship material, making helium a safer choice despite being heavier.

11. What is the most probable speed in the Maxwell Distribution?

The most probable speed is the speed at which the Maxwell Distribution curve reaches its peak. It represents the speed that the largest number of molecules in the gas are likely to have at a given temperature.

12. How does molecular mass affect the Maxwell Distribution?

For gases at the same temperature, molecules with lower mass will have a broader distribution curve with a peak at higher speeds, while heavier molecules will have a narrower distribution curve with a peak at lower speeds.

13. How does the Maxwell Distribution relate to the concept of mean free path?

The mean free path, which is the average distance a molecule travels between collisions, is influenced by the Maxwell Distribution. Faster molecules will typically have longer mean free paths, while slower molecules have shorter ones.

14. How does the Maxwell Distribution contribute to our understanding of chemical reactions?

The Maxwell Distribution is crucial in understanding reaction rates. Only molecules with sufficient kinetic energy (speed) can overcome the activation energy barrier to react. The distribution helps predict the fraction of molecules that have enough energy for a reaction at a given temperature.

15. What is the relationship between the Maxwell Distribution and the ideal gas law?

The Maxwell Distribution is consistent with the ideal gas law. The average kinetic energy derived from the distribution is directly related to temperature, which is a key parameter in the ideal gas law. Both are fundamental to the kinetic theory of gases.

16. What is the Maxwell Distribution Law?

The Maxwell Distribution Law describes the distribution of molecular speeds in an ideal gas at thermal equilibrium. It states that in a gas at a given temperature, molecules have a range of speeds, with most molecules moving at speeds close to the most probable speed, while fewer molecules move at very high or very low speeds.

17. Can the Maxwell Distribution be applied to liquids or solids?

The Maxwell Distribution is specifically derived for ideal gases. While it can provide some insights into the behavior of particles in liquids and solids, it's not directly applicable due to the strong intermolecular forces present in these states of matter.

18. What assumptions are made in deriving the Maxwell Distribution?

The key assumptions include: the gas is ideal, molecules move randomly, collisions are elastic, and the gas is in thermal equilibrium. It also assumes that classical mechanics apply, which limits its applicability at very low temperatures or for very light particles.

19. What is the significance of the tail of the Maxwell Distribution?

The tail of the Maxwell Distribution represents the small fraction of molecules moving at very high speeds. These high-speed molecules are important in understanding phenomena like evaporation, where only the fastest molecules can escape the liquid surface.

20. How does pressure affect the Maxwell Distribution?

Pressure alone does not affect the Maxwell Distribution. The distribution depends only on temperature and the type of gas (molecular mass). However, in a real gas, extremely high pressures can lead to deviations from ideal gas behavior, indirectly affecting the speed distribution.

21. Can the Maxwell Distribution predict the number of molecules within a certain speed range?

Yes, the area under the Maxwell Distribution curve between two speed values gives the fraction of molecules with speeds in that range. This allows for quantitative predictions about molecular speeds in a gas.

22. How does the Maxwell Distribution relate to the equipartition theorem?

The Maxwell Distribution is consistent with the equipartition theorem, which states that energy is equally distributed among all degrees of freedom in a system at thermal equilibrium. This theorem helps explain why the average kinetic energy per molecule is proportional to temperature.

23. What is the relationship between the Maxwell Distribution and the Boltzmann Distribution?

The Maxwell Distribution is a specific case of the more general Boltzmann Distribution. While the Maxwell Distribution describes the distribution of molecular speeds in an ideal gas, the Boltzmann Distribution describes the distribution of particles among energy states in any system at thermal equilibrium.

24. Can the Maxwell Distribution be experimentally verified?

Yes, the Maxwell Distribution has been experimentally verified using various techniques. One common method is molecular beam experiments, where the speeds of molecules in a beam are measured and compared to the predicted distribution.

25. What is the significance of the Maxwell Distribution in atmospheric science?

In atmospheric science, the Maxwell Distribution helps explain phenomena like the escape of light gases from a planet's atmosphere. It also aids in understanding the vertical distribution of gases in the atmosphere and their interactions with solar radiation.

26. How does the Maxwell Distribution explain the phenomenon of effusion?

Effusion, the process by which gases escape through small holes, is explained by the Maxwell Distribution. Lighter gases have a higher proportion of fast-moving molecules according to the distribution, which is why they effuse more quickly than heavier gases.

27. Can the Maxwell Distribution be applied to a mixture of gases?

Yes, in a mixture of ideal gases, each gas component follows its own Maxwell Distribution based on its molecular mass. The overall behavior of the mixture can be understood by considering the individual distributions of each component.

28. How does the Maxwell Distribution relate to the concept of partial pressure in gas mixtures?

In a gas mixture, each component contributes its partial pressure based on its concentration and follows its own Maxwell Distribution. The total pressure is the sum of these partial pressures, consistent with Dalton's law of partial pressures.

29. How does the Maxwell Distribution account for the temperature independence of gas viscosity?

The Maxwell Distribution shows that as temperature increases, molecules move faster but collide more frequently. These effects tend to cancel out, explaining why gas viscosity is relatively independent of temperature, a phenomenon not immediately obvious without statistical mechanics.

30. How does the Maxwell Distribution relate to the concept of degrees of freedom in molecules?

The Maxwell Distribution is derived assuming three translational degrees of freedom for monatomic gases. For more complex molecules with rotational and vibrational degrees of freedom, the distribution needs to be modified, but the basic principles remain the same.

31. Can the Maxwell Distribution explain why some gases deviate from ideal behavior?

While the Maxwell Distribution itself assumes ideal gas behavior, deviations from the distribution at very high pressures or low temperatures can help explain non-ideal gas behavior. These deviations occur when intermolecular forces become significant.

32. What is the significance of the Maxwell Distribution in plasma physics?

In plasma physics, the Maxwell Distribution is often used to describe the velocity distribution of charged particles. However, in many plasma conditions, deviations from the Maxwell Distribution occur, leading to the study of non-Maxwellian distributions.

33. How does the Maxwell Distribution contribute to our understanding of the greenhouse effect?

The Maxwell Distribution helps explain how greenhouse gases interact with infrared radiation. The distribution of molecular speeds and energies determines which vibrational and rotational energy states are populated, affecting the absorption and emission of infrared radiation.

34. Can the Maxwell Distribution be used to explain the behavior of electrons in metals?

While the Maxwell Distribution was originally derived for gases, a similar distribution (the Fermi-Dirac distribution) describes the energy distribution of electrons in metals. The Maxwell Distribution serves as a classical limit of this quantum mechanical distribution at high temperatures.

35. How does the Maxwell Distribution relate to the concept of entropy in statistical mechanics?

The Maxwell Distribution represents the most probable distribution of molecular speeds in a gas at equilibrium, which corresponds to the state of maximum entropy. This connection helps link microscopic molecular behavior to macroscopic thermodynamic properties.

36. Can the Maxwell Distribution explain why evaporative cooling is more effective in dry climates?

Yes, the Maxwell Distribution shows that only the fastest-moving molecules can overcome intermolecular forces and evaporate. In dry climates, more of these high-energy molecules can escape, leading to more effective cooling as they carry away thermal energy.

37. How does the Maxwell Distribution relate to the concept of effusive flow in vacuum systems?

Effusive flow, where gas molecules pass through an orifice smaller than their mean free path, is directly related to the Maxwell Distribution. The distribution determines the rate at which molecules with sufficient energy to escape will encounter the orifice.

38. What is the significance of the Maxwell Distribution in understanding gas-surface interactions?

The Maxwell Distribution helps explain phenomena like adsorption and desorption at gas-surface interfaces. The energy distribution of gas molecules determines their ability to stick to surfaces or overcome surface binding energies and desorb.

39. What role does the Maxwell Distribution play in understanding the behavior of rarefied gases?

In rarefied gases, where the mean free path is comparable to the container size, the Maxwell Distribution remains important but needs to be considered alongside boundary effects. It helps explain phenomena like slip flow and temperature jump at surfaces.

40. How does the Maxwell Distribution relate to the concept of thermal de Broglie wavelength?

The thermal de Broglie wavelength, which is important in quantum mechanics, is related to the most probable speed from the Maxwell Distribution. This connection helps determine when quantum effects become significant for a gas at a given temperature.

41. What is the relationship between the Maxwell Distribution and the concept of thermal conductivity in gases?

Thermal conductivity in gases is related to the transfer of kinetic energy between molecules, which is described by the Maxwell Distribution. The distribution helps explain why thermal conductivity in gases increases with temperature, unlike in most solids.

42. How does the Maxwell Distribution explain the phenomenon of thermal transpiration?

Thermal transpiration, where gas flows from a cold to a hot region through a porous barrier, can be explained using the Maxwell Distribution. The distribution shows that molecules on the hot side have higher average speeds, leading to a net flow towards the cold side to maintain pressure equilibrium.

43. What role does the Maxwell Distribution play in understanding the escape of atmospheric gases from planets?

The Maxwell Distribution helps explain atmospheric escape by showing that a small fraction of gas molecules always have speeds exceeding the escape velocity. Lighter gases have more high-speed molecules, explaining why planets retain heavier gases more easily than lighter ones.

44. How does the Maxwell Distribution contribute to our understanding of gas centrifuges used in isotope separation?

Gas centrifuges exploit the slight differences in the Maxwell Distributions of different isotopes. Heavier isotopes have a narrower speed distribution with a lower average speed, allowing for separation when subjected to centrifugal forces.

45. How does the Maxwell Distribution contribute to our understanding of gas chromatography?

In gas chromatography, the Maxwell Distribution helps explain the movement and separation of different molecular species. The distribution of molecular speeds affects how different molecules interact with the stationary phase and move through the column.

46. Can the Maxwell Distribution be used to explain the phenomenon of Knudsen diffusion in porous materials?

Yes, Knudsen diffusion, which occurs when the pore size is smaller than the mean free path of gas molecules, can be understood using the Maxwell Distribution. The distribution of molecular speeds determines how molecules interact with pore walls and diffuse through the material.

47. How does the Maxwell Distribution relate to the concept of molecular beams in experimental physics?

Molecular beam experiments often use the Maxwell Distribution to interpret results. The distribution helps predict the velocity and energy distribution of molecules in the beam, which is crucial for understanding collision experiments and spectroscopic measurements.

48. How does the Maxwell Distribution contribute to our understanding of gas-liquid equilibrium?

The Maxwell Distribution helps explain the dynamic equilibrium between a liquid and its vapor. It describes the speed distribution of molecules in the vapor phase and helps determine the rate at which molecules can overcome surface tension and evaporate.

49. Can the Maxwell Distribution be used to explain why certain gases are more easily liquefied than others?

Yes, the Maxwell Distribution shows that heavier gases have a narrower speed distribution with lower average speeds. This means they have less kinetic energy on average, making it easier to overcome their kinetic energy with intermolecular attractions and liquefy them.

50. How does the Maxwell Distribution relate to the concept of mean molecular speed in statistical thermodynamics?

The mean molecular speed is directly calculated from the Maxwell Distribution. It represents the average speed of all molecules in the gas and is a key parameter in relating microscopic molecular motion to macroscopic properties like pressure and temperature in statistical thermodynamics.