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Introduction

The kinetic molecular theory is used to determine the motion of a molecule of an ideal gas under a certain set of conditions. However, when looking 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 velocity, for example, the Maxwell-Boltzmann distribution is given by

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs  (1.1)

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.1

Additionally, the function can 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

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs (1.2)

Finally, the Maxwell-Boltzmann distribution can be used to determine the distribution of the kinetic energy of for a set of molecules. The distribution of the kinetic energy is identical to the distribution of the speeds for a certain gas at any temperature.2 

Plotting the Maxwell-Boltzmann Distribution Function

Figure 1 shows the Maxwell-Boltzmann distribution of speeds for a certain gas at a certain temperature, such as nitrogen at 298 K. The speed at the top of the curve is called the most probable speed because the largest number of molecules have that speed.

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs

Figure 1: The Maxwell-Boltzmann distribution is shifted to higher speeds and is broadened at higher temperatures.

Figure 2 shows how the Maxwell-Boltzmann distribution is affected by temperature. At lower temperatures, the molecules have less energy. Therefore, the speeds of the molecules are lower and the distribution has a smaller range. As the temperature of the molecules increases, the distribution flattens out. Because the molecules have greater energy at higher temperature, the molecules are moving faster. 

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs

Figure 2: The Maxwell-Boltzmann distribution is shifted to higher speeds and is broadened at higher temperatures. 

Figure 3 shows the dependence of the Maxwell-Boltzmann distribution on molecule mass. On average, heavier molecules move more slowly than lighter molecules. Therefore, heavier molecules will have a smaller speed distribution, while lighter molecules will have a speed distribution that is more spread out. 

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs

Figure 3: The speed probability density functions of the speeds of a few noble gases at a temperature of 298.15 K (25 °C). The y-axis is in s/m so that the area under any section of the curve (which represents the probability of the speed being in that range) is dimensionless.

Related Speed Expressions

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

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs (1.3)

which is

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs (1.4)

The average speed is the sum of the speeds of all the molecules divided by the number of molecules.

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs (1.5)

The root-mean-square speed is square root of the average speed-squared. 

Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs (1.6)

where

  • RR is the gas constant,
  • TT is the absolute temperature and
  • MM is the molar mass of the gas.

It always follows that for gases that follow the Maxwell-Boltzmann distribution (if thermallized)

Cmp<Cavg<Crms          (1.7)

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FAQs on Boltzmann Distribution - Statistical Thermodynamics, Physical Chemistry, CSIR-NET - Government Jobs

1. What is the Boltzmann distribution in statistical thermodynamics?
Ans. The Boltzmann distribution is a probability distribution that describes the distribution of particles in different energy states in a system at thermal equilibrium. It provides the probability of finding a particle in a specific energy state, and it is derived from the principles of statistical thermodynamics.
2. How is the Boltzmann distribution derived in statistical thermodynamics?
Ans. The Boltzmann distribution is derived by considering the probability of a particle occupying a particular energy state in a system at thermal equilibrium. It is derived using the principles of statistical mechanics, such as the concept of microstates and macrostates, and the assumption that the system is in equilibrium. The distribution is obtained by maximizing the entropy of the system under the constraints of fixed energy and fixed number of particles.
3. What is the significance of the Boltzmann distribution in physical chemistry?
Ans. The Boltzmann distribution is of great significance in physical chemistry as it provides a mathematical framework to describe the distribution of particles in different energy states in a system at thermal equilibrium. It is used to calculate probabilities and average values of various thermodynamic properties, such as energy, entropy, and free energy. The Boltzmann distribution also helps in understanding the behavior of gases, liquids, and solids at different temperatures.
4. How is the Boltzmann distribution related to CSIR-NET exam in physical chemistry?
Ans. The Boltzmann distribution is an important topic in physical chemistry, which is a part of the syllabus for the CSIR-NET exam. Candidates appearing for this exam are expected to have a thorough understanding of statistical thermodynamics, including the Boltzmann distribution. Questions related to the derivation of the distribution, its applications, and its significance in physical chemistry can be asked in the exam.
5. Are there any government job opportunities for candidates with knowledge of the Boltzmann distribution and statistical thermodynamics?
Ans. Yes, candidates with knowledge of the Boltzmann distribution and statistical thermodynamics can explore government job opportunities in various fields. These include research positions in government research institutes, teaching positions in government universities and colleges, and scientific positions in government laboratories. Additionally, candidates can also apply for jobs in industries related to energy, materials, and chemical processes where thermodynamic principles are applied.
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