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The Boltzmann equation or Boltzmann transport equation (BTE) represents the statistical behavior of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in the year 1872.
- He proposed the modern concept of entropy in 1877, which is defined as a measure of a system's statistical disorder.
- The Boltzmann constant was named after Max Planck.
- A fluid having temperature gradients in space causes heat to move from hotter to cooler places due to the random yet biased motion of the particles that make up that fluid.
In current literature, the phrase Boltzmann equation refers to any kinetic equation that represents the change of a macroscopic variable in a thermodynamic system, such as energy, charge, or particle number, in a more generic meaning.
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Key Terms: Boltzmann equation, Boltzmann transport equation, Momentum, Position, Entropy, Thermodynamic system, Energy, State of equilibrium, Thermodynamics
What is Boltzmann Equation?
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The Boltzmann equation is not derived by analyzing each particle's specific positions and momenta in the fluid. Instead, it considers a probability distribution for the position of momentum of a particle.
- This refers to the probability that the particle will occupy a certain very small region of space.
- Mathematically, the volume of the element d3r centered at the point r and with momentum approximately equal to a given momentum vector at a given instant in time.
- The Boltzmann equation can be used to calculate how physical quantities, such as heat energy and momentum, change as a fluid is transported.
- The Boltzmann equation is a nonlinear integrodifferential equation in which the unknown function is the probability density function in the six-dimensional space of a particle's position and momentum.
The below given is the Boltzmann equation:
| \(\frac{P_{Sb}}{P_{Sa}}=\frac{N_{b}}{N_{a}}=\frac{gbe^-{\frac{E_b}{kT}}}{gae^-\frac{E_a}{kT}}=\frac{gb}{ga}e^-\frac{(E_b-E_a)}{kT}\) |
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Ludwig Boltzmann
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Ludwig Boltzmann was a popular physicist and philosopher in the 19th century.- He discovered the statistical explanation for the second law of thermodynamics and contributed to the growth of statistical mechanics.
- Boltzmann's kinetic theory of gas was also an important contribution.
- The Boltzmann equation is commonly known as the Boltzmann transport equation (BTE).
- It is used to study the statistical behavior of a thermodynamic system that is not in a state of equilibrium. Ludwig Boltzmann developed it in 1872.
Applications of Boltzmann Equation
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The application of the Boltzmann equation is:
- Conservation equation: The Boltzmann equation is used in fluid dynamics to derive mass, momentum, charge, and energy conservation laws.
- Hamiltonian Mechanics: With the use of several mathematical formulations, classical mechanics was transformed into Hamiltonian mechanics.
- Quantum theory and violation of particle number: In collisions, the quantum Boltzmann equation is employed to calculate the number of non-conserved particles. Physical cosmology makes extensive use of these.
- General relativity and astronomy: The Boltzmann equation is used in galactic dynamics.
Solved Example
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Ques. Find out the temperature at which the number of hydrogen atoms is equal in the ground state when n = 1 and the second excited state is n = 3. Also, the required energy is E3 = -1.5 eV.
Ans. Given,
- Ground state, n = 1
- Excited-state, n = 3
- The required energy is E3 = -1.5 eV
By using the Boltzmann equation,
N3/N1 = 1 = 18/2 e-12.1/kT
⇒ kT = -(12.1/ln 0.111) = 5.51 eV
⇒ T = 5.51 x 40 x 300 = 66,000 K
But we know that for N1 = N2,
Therefore, the required temperature is 85,400 K
For this temperature, kT = 7.12 eV
Hence,
N3 = N1 18/2 e-12.1/7.12
⇒ N3 = 1.64 N1
Things to Remember
- The Boltzmann equation represents the statistical behavior of a thermodynamic system not in a state of equilibrium.
- It is also known as the Boltzmann transport equation (BTE).
- The Boltzmann constant was named after Max Planck.
- The Boltzmann equation can be used to calculate how physical quantities, such as heat energy and momentum, change as a fluid is transported.
- The Maxwell-Boltzmann distribution for molecular speeds in gas was one of Boltzmann's most significant scientific discoveries.
- The Boltzmann distribution over energies and Maxwell-Boltzmann statistics remain the cornerstones of classical statistical mechanics.
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Sample Questions
Ques. Who put forth the Boltzmann equation? (1 Mark)
Ans.
In 1872, the famous scientist and philosopher Ludwig Boltzmann proposed the Boltzmann equation.Ques. What is the Boltzmann equation also known as? (1 Mark)
Ans. The Boltzmann equation is also known as the Boltzmann transport equation.
Ques. What are the applications of the Boltzmann equation? (2 Marks)
Ans. The applications of the Boltzmann equation are
- Derivation of conservation equations
- Quantum theory and violation of particle number
- Hamiltonian mechanics
- General relativity and astronomy
Ques. Give the Boltzmann equation. (2 Marks)
Ans. The Boltzmann equation is given by
\(\frac{P_{Sb}}{P_{Sa}}=\frac{N_{b}}{N_{a}}=\frac{gbe^-{\frac{Eb}{kT}}}{ga^-\frac{Ea}{kT}}=\frac{gb}{ga}e^-\frac{(Eb-Ea)}{kT}\)
Ques. The Boltzmann equation is used in the derivation of conservation laws of mass. is it true or false? (1 Mark)
Ans. Yes, it is true that the Boltzmann equation is used in the derivation of conservation laws of mass.
Ques. What is the SI unit entropy? (1 Mark)
Ans. The SI unit of Entropy is finally given as Joule/Kelvin.
Ques. What is the unit of enthalpy? (1 Mark)
Ans. In the International System of Units (SI), the unit of measurement for enthalpy is the joule.
Ques. What is the conservation of mass? (2 Marks)
Ans. The law of conservation of mass states that mass is neither created nor destroyed in a chemical reaction.
Ques. What is hydrogen's ionization energy? Provide the number as well as a brief explanation. (2 Marks)
Ans. As stated in the problem, the ionization energy of hydrogen is the energy required to remove the electron from the ground state - effectively a transition from n=1 to n=inf, which simply corresponds to the energy of the ground state: E1=χi=13.6 eV.
Ques. At T =10000 K, do you think a large number of hydrogen atoms will be ionized? Why do you think that is? (2 Marks)
Ans. Only half of the atoms have been heated to n=2 according to the Boltzmann equation at T=85,000 K. (requiring energy of 10.2 eV). Ionizing hydrogen, on the other hand, necessitates significantly more energy (13.6 eV). Consequently, based solely on the answers for a) and b) a significant number of H atoms would not be expected to be ionized at T=10,000 K.
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