Maxwell-Boltzmann distribution

Examples of Maxwell-Boltzmann distribution in the following topics:

• Speed Distribution of Molecules

  • A gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution.
  • Maxwell-Boltzmann distribution is a probability distribution.
  • The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above.
  • The Maxwell-Boltzmann distribution of molecular speeds in an ideal gas.
  • Describe the shape and temperature dependence of the Maxwell-Boltzmann distribution curve

• Distribution of Molecular Speeds and Collision Frequency

  • The Maxwell-Boltzmann Distribution describes the average molecular speeds for a collection of gas particles at a given temperature.
  • In theory, this energy can be distributed among the gaseous particles in many ways, and the distribution constantly changes as the particles collide with each other and with their boundaries.
  • This results in an asymmetric curve, known as the Maxwell-Boltzmann distribution.
  • Velocity distributions are dependent on the temperature and mass of the particles.
  • When we plot this, we see that an increase in temperature causes the Boltzmann plot to spread out, with the relative maximum shifting to the right.

• Overview of Temperature and Kinetic Theory

  • The distribution of the speeds (which determine the translational kinetic energies) of the particles in a classical ideal gas is called the Maxwell-Boltzmann distribution.
  • (k: Boltzmann's constant).

• Stastical Interpretation of Entropy

  • The most likely conditions (or macrostate) for the gas are those we see all the time—a random distribution of atoms in space with a Maxwell-Boltzmann distribution of speeds in random directions, as predicted by kinetic theory as shown in (a).
  • (a) The ordinary state of gas in a container is a disorderly, random distribution of atoms or molecules with a Maxwell-Boltzmann distribution of speeds.

• The Arrhenius Equation

  • The equation combines the concepts of activation energy and the Boltzmann distribution law into one of the most important relationships in physical chemistry:
  • Recall that the exponential part of the Arrhenius equation ($e^{\frac{-E_a}{RT}}$) expresses the fraction of reactant molecules that possess enough kinetic energy to react, as governed by the Maxwell-Boltzmann distribution.

• Order to Disorder

  • This notion was initially postulated by Ludwig Boltzmann in the 1800s.
  • Rather than having two masses at different temperatures and with different distributions of molecular speeds, we now have a single mass with a uniform temperature.

• Variation of Pressure With Depth

  • Since, for gases and liquids, the force acting on a system contributing to pressure does not act on a specific point or particular surface, but rather as a distribution of force, analyzing pressure as a measure of energy per unit volume is more appropriate.
  • Thus the force contributing to the pressure of a gas within the medium is not a continuous distribution as for liquids and the barometric equation given in must be utilized to determine the pressure exerted by the gas at a certain depth (or height) within the gas (p0 is the pressure at h = 0, M is the mass of a single molecule of gas, g is the acceleration due to gravity, k is the Boltzmann constant, T is the temperature of the gas, and h is the height or depth within the gas).
  • The force contributing to the pressure of a gas within the medium is not a continuous distribution as for liquids and the barometric equation given in this figure must be utilized to determine the pressure exerted by the gas at a certain depth (or height) within the gas (p0 is the pressure at h = 0, M is the mass of a single molecule of gas, g is the acceleration due to gravity, k is the Boltzmann constant, T is the temperature of the gas, and h is the height or depth within the gas)

• Particle-Wave Duality

  • In 1861, James Clerk Maxwell explained light as the propagation of electromagnetic waves according to the Maxwell's equations.
  • The standard interpretation is that the act of measurement causes the set of probabilities, governed by a probability distribution function acquired from a "wave", to immediately and randomly assume one of the possible values, leading to a "particle"-like result.

• Radiation

  • The radiated energy depends on its intensity, which is represented by the height of the distribution .
  • The rate of heat transfer by emitted radiation is determined by the Stefan-Boltzmann law of radiation:
  • where =5.67×10−8 J s-1⋅m-2⋅K-4 is the Stefan-Boltzmann constant, A is the surface area of the object, and T is its absolute temperature in kelvin.

• The Importance of Sociocultural Differences

  • Maxwell House advertised itself as the "great American coffee" in Germany.