The most probable distribution of energy among particles in a gas at constant temperature and volume is given by the Maxwell-Boltzmann distribution. This distribution describes the probability of a particle having a certain energy or speed in an ideal gas. The Maxwell-Boltzmann distribution is derived using the principles of statistical mechanics and thermodynamic ensembles.Statistical mechanics is a branch of physics that uses statistical methods to explain the behavior of a large number of particles in a system. It provides a connection between the microscopic properties of individual particles and the macroscopic properties of the system, such as temperature and pressure.Thermodynamic ensembles are collections of a large number of microstates possible arrangements of particles that are consistent with certain macroscopic properties, such as energy, temperature, and volume. The most common ensembles are the microcanonical ensemble constant energy , canonical ensemble constant temperature , and grand canonical ensemble constant temperature and chemical potential .In the case of a gas at constant temperature and volume, the canonical ensemble is the most appropriate. The canonical ensemble is described by the Boltzmann distribution, which gives the probability of a system being in a certain microstate based on its energy and the system's temperature. The Boltzmann distribution is given by:P E = 1/Z * e^-E/kT where P E is the probability of a microstate with energy E, k is the Boltzmann constant, T is the temperature, and Z is the partition function, which is a normalization factor ensuring that the sum of probabilities of all microstates is equal to 1.The Maxwell-Boltzmann distribution is derived from the Boltzmann distribution by considering the energy of particles in an ideal gas. The energy of a particle in an ideal gas is given by its kinetic energy:E = 1/2 * m * v^2where m is the mass of the particle and v is its speed. By substituting this expression for energy into the Boltzmann distribution and considering the distribution of speeds in three dimensions, we obtain the Maxwell-Boltzmann distribution for the speed of particles in an ideal gas:f v = m / 2 * * k * T ^3/2 * 4 * * v^2 * e^-m * v^2 / 2 * k * T The Maxwell-Boltzmann distribution shows that the probability of a particle having a certain speed depends on its mass, the temperature of the system, and the Boltzmann constant. At a given temperature, particles with lower mass will have a higher probability of having higher speeds, while particles with higher mass will have a higher probability of having lower speeds.In summary, the most probable distribution of energy among particles in a gas at constant temperature and volume is given by the Maxwell-Boltzmann distribution, which is derived using the principles of statistical mechanics and thermodynamic ensembles, specifically the canonical ensemble and the Boltzmann distribution.