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Maxwell Boltzmann distribution is used to study the distribution of the speed of gas (particularly ideal gas) molecules at a certain temperature. This is because, in air, the movement of molecules occurs at varying speeds. Certain molecules travel at faster speeds as compared to others and some do not move in the air at all. Furthermore, it is not possible to determine the speed of a single gas molecule. Thus, the distribution of speed of the molecules in gas at a certain temperature is calculated using Maxwell Boltzmann distribution formula.
Read More: Concepts in Chemistry
Key Takeaways: Ideal Gas, Kinetic Theory of Gases, Distribution Function, Related Speed Expressions
What is Maxwell Boltzmann Distribution
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Maxwell Boltzmann distribution addresses the speed of the molecules in the gases (preferably ideal gases) where they move inside a container. Furthermore, it also assumes that the gas molecules do not demonstrate any interaction with each other barring for collisions.
Thus, this distribution is the probability distribution which is used to describe the velocities of different particles present inside a closed system at a particular temperature.
Maxwell Boltzmann distribution shifts to higher speeds and broadens at higher temperatures
The speed probability density functions for the speeds
The distribution was first derived by Maxwell in the 1860s and later by Boltzmann, in the 1870s, by carrying out significant investigations into the physical origins of this distribution. Furthermore, the distribution can be derived on the ground that it maximises the entropy of the system.
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| Value of Boltzmann Constant | Atoms and Molecules | Velocity |
Ideal Gas and Ideal Gas law
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In an ideal gas, the molecules of the gas are assumed to collide in somewhat perfect elastic collisions. Thus resulting in a constant energy through the system even after collisions. Furthermore, it is also assumed that the molecules of the gas are significantly far apart which means that they don’t face space crunch and can be considered as point particles.
Ideal Gases and Non Ideal Gases
Such types of ideal gases are neither too cold nor too hot, which makes them not undergo either ionisation or have any other quantum effects. All these properties make ideal gases easiest to analyse.
Ideal Gas Law
Ideal gas law is used to relate the volume, pressure as well as the temperature of an ideal gas. The pressure P of the gas represents the force per unit area which it exerts on the walls of the container it is put in. Its SI unit is Pascal (Pa) wherein 1Pa = 1N/m2. The volume V represents the amount of space which is taken up by the gas. Its SI unit is m3. The temperature of the gas T represents the average kinetic energy per molecule. Its SI unit is Kelvin.
Ideal Gas Law
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| Pressure of an Ideal Gas | Charles law | Avogadro’s law |
What is Kinetic theory of Gases
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- In an ideal gas, instead of taking into consideration the exact physics of each molecule of the gas, they can be treated as a function of their motions which are relatively random. As a consequence, statistics can be applied in order to understand what exactly is going on in the system.
- Physicists James Clerk Maxwell and Ludwig Boltzmann, in the 19th century, gave kinetic theory of gases which is based on the simplified idea described above.
Kinetic Theory of Gases
- Each molecule of the gas has kinetic energy but not every molecule in the gas will have the same kinetic energy as a result of the constant collision of the molecules.
- Thus, the exact distribution of the energies of the molecules in a container with a fixed temperature is given by Maxwell Boltzmann distribution.
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Distribution of Velocity Vector
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Kinetic molecular theory is used in order to determine the motion of a molecule of an ideal gas under a certain set of conditions. However, at a mole of ideal gas, it is impossible to measure the velocity of each molecule at every instant of time. Therefore, the Maxwell Boltzmann distribution is used to determine how many molecules are moving between velocities v and v + dv.
Assuming that the one-dimensional distributions are independent of one another, that the velocity in the y and z directions does not affect the x velocity, the Maxwell Boltzmann distribution is given by
Where;
- dN/N is the fraction of molecules moving at velocity v to v + dv,
- m is the mass of the molecule,
- kb is the Boltzmann constant, and
- T is the absolute temperature.
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| Dalton’s law of partial pressure | Fluid Friction | Boyle’s law |
Distribution Function
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The function can also be written in terms of the scalar quantity speed c instead of the vector quantity velocity. This form of the function defines the distribution of the gas molecules moving at different speeds, between c1 and c2, thus the equation can be given as
where
m is the particle mass,
k is the Boltzmann's constant, and
T thermodynamic temperature.
Moreover, the Maxwell-Boltzmann distribution can be used to determine the distribution of the kinetic energy for a set of molecules. This is because the distribution of the kinetic energy is identical to the distribution of the speeds for a certain gas at any temperature.
Read Also: Gas Constant
Distribution of Energy
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In terms of distribution of the particles on the basis of amount of amount of energy between identical but somewhat distinguishable particles, the Maxwell Boltzmann distribution can be written as :
1 / Exponential(energy/(Boltzmann constant Temperature))
Or
f = 1/exp(-E/kT)
Where:
f = Energy distribution
E = energy of the system
k = Boltzmann constant. (1.38*10-²³ m² kg /(s K²))
T = Absolute Temperature in Kelvin.
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Related Speed Expressions
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Three speed expressions can be derived from the Maxwell-Boltzmann distribution: the most probable speed, the average speed, and the root-mean-square speed. The most probable speed is the maximum value on the distribution plot. This is established by finding the velocity when the following derivative is zero.
which is
Most probable speed equation
The average speed is the sum of the speeds of all the molecules divided by the number of molecules.
The root-mean-square speed is the square root of the average speed-squared.
where,
- R is the gas constant,
- T is the absolute temperature and
- M is the molar mass of the gas.
It always follows that for gases that follow the Maxwell-Boltzmann distribution (if thermalized)
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Things to Remember
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- Maxwell Boltzmann law is given by the expression f = 1/exp(-E/kT) where k is the Boltzmann constant with value 1.98 X 10-²³ J/K and T is the Temperature. The constant α depends on the volume and the temperature of the gas.
- In kinematic theory of gases, macroscopic quantities (such as press and temperature) are explained by considering microscopic (random) motion of molecules.
- Maxwell Boltzmann distribution is employed in order to study the distribution of speed of different molecules at a particular temperature.
- The theory was given by scientists James Maxwell and Ludwig Boltmann.
- Maxwell Boltzmann distribution equation is predominantly utilised to determine the speed of the molecules distributed in an ideal gas.
- The equation helps in giving information regarding the occurrence of a particle at a particular temperature as well as given energy.
- In a mathematical sense, the Maxwell Boltzmann distribution is a type of chi distribution containing three degrees of freedom with scale parameters used to measure the speeds of the particles in the gas in units which are proportional to the square root of T/m (ratio of temperature over mass).
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Sample Questions
Ques: In case the temperature of a black body radiator is 5000000 K, find out the value of its distribution at that state at an energy of 1x10 (-19) J. [2 marks]
Ans. The distribution can be found out by using the formula;
f = 1/exp(-E/kT)
Substituting the values,
f = 1/exp (-1*10(-19) J/(1.38*10(-23) m2 Kg/(s K2)*5000000 K))
Thus, the value of its distribution is 0.9998.
Ques: At -23°C, what will be the Kinetic energy of 2g of oxygen? [2 marks]
Ans: Kinetic energy(KE) = 3nRT/2
Here, T = 250 K and R = 8.314 J/K
n = 2/32 = 1/16 mol
Therefore,
KE = 3 X 8.314 X 250/32
KE = 194.86 J
Ques: At 27°C, what will be the rms speed of methane gas? [2 marks]
Ans: As we know,
\((v_{rms} = \sqrt{\frac{3kT}{M}})\)Here, T = 300 K, k = 1.38 X 10-23 J/K, M = 16/6.022 X 10-26 = 2.65 X 10-26
Therefore,
\((v_{rms} = \sqrt{\frac{3X 30 X 1.38}{2.65}})\)Thus, the rms speed of methane will be 678 m/s at 27°C .
Ques: In case the temperature of a black body radiator is 5000K, find out the value of its distribution at that state at an energy of 1x10 (-19) J. [2 marks]
Ans. The distribution can be found out by using the formula;
f = 1/exp(-E/kT)
Substituting the given values in the formula,
f = 1/exp (-1*10(-19) J/(1.38*10(-23) m2 Kg/(s K2)*5000 K))
Thus, the value of its distribution is 0.234
Ques: What would be the most probable velocity for one molecule of oxygen at the temperature of 300K? [3 marks]
Ans: Mass of one oxygen molecule = 32 g/mol
= 32 X 10-3/ 6.022 X 1023
= 5.31 X 10-26 Kg
Now,
\((V_p = \sqrt{\frac{2kT}{M}})\)Here, T = 300 K, k = 1.98 X 10^23 and M = 32
\((V_p = \sqrt{\frac{2X 1.98X 30}{5.31}} \times 10^4\)Therefore, the velocity of the oxygen molecule would be 714 m/s.
Ques: Calculate pressure of P of ideal gas with concentration n and temperature T using Maxwell’s distribution. [3 marks]
Ans.
Ques: What will be the average speed of hydrogen molecules at a temperature of 300K? [3 marks]
Ans:
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