Boltzmann Distribution Boltzmann Distribution The Boltzmann distribution (also called the Gibbs Distribution[1]) is a certain distribution function or probability measure for the distribution of the states of a system. The distribution was discovered in the context of classical statistical mechanics by J.W. Gibbs in 1901. It underpins the concept of the canonical ensemble, providing the underyling distribution. A special case of the Boltzmann distribution, used for describing the velocities of particles of a gas, is the Maxwell–Boltzmann distribution. In more general mathematical settings, the Boltzmann distribution is also known as the Gibbs measure. Boltzmann Distribution Equation In according to kinetic molecular theory, pressure is generated when the gas particles collide the walls of the container in an elastic manner. 1. Let us assume a cubical container of volume V = L3 containing N molecules. 2. Let us assume that a particle colliding on x coordinate with momentum p 3. The change in momentum is Δp= Pf-Pi = Px-(-Px) = 2Px = 2mvx (because the collision is perfectly elastic the momentum is not lost)
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4. The change in time Δt = 2L/vx, which is the time for one specific collision. Here, L is the length of cube. 5. So the force due to this collision is equal to Δp/Δt = 2mvx2/2L = mvx2/L. The total force given by N molecules is Nmvx2/L. 6. But there are three co-ordinates and the average velocity of one co-ordinate is equal to 1/3rd of average velocity of the total particles in the system or Vx= V/3. Changing this in the force formula we will get Force = Nmv2/3L. 7. Hence the pressure of gas= force / area = Nmv2/3L÷ L2 = Nmv2/3L3 = Nmv2/3V. Here N/V= number of moles in the system = n so pressure P = nmv2/3 Kinetic Molecular Theory of Gases 1. Gases are made up of small tiny particles. This particles move in a random passion continuously. 2. There is no attractive force in between gas particles and their movement is random without the influence of the other. 3. The number of particles is very large and hence we can apply the statistical treatment like Boltzmann distribution. 4. The volume occupied by the particles added up all together is negligible comparing with volume of the container. 5. During their movement, the particles collide with each other as well as with the volume of the container. This collision is perfectly elastic.
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6. Pressure is generated due to the collision of gas particles with the wall of the container. 3. Relationship between volume and temperature (Charles Law) When we increase the temperature, the kinetic energy of particles will increase as KE is directly related with temperature. KE = 3/2RT Hence the gas particles will strike the wall of container with more pressure. But the wall of container is flexible and can expand. So the volume of the container will increase with increase in temperature. This proves Charles Law. VÎąT 4. Dalton's Law of partial pressure When there are different gas particles in a container each will strike the wall of container of its own without the influence of the other particles. Hence the total pressure of the system is equal to the sum of pressure generated by each type of particles when they are alone in a system. P = P1 + P2 + P3 This explains the Dalton's law of partial pressure.
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