To calculate the probability of finding a system composed of two ideal gas particles in a given volume, temperature, and energy range, we will use statistical mechanics and the canonical ensemble. The canonical ensemble is a statistical ensemble representing all possible states of a system in thermal equilibrium with a heat reservoir at a fixed temperature.First, let's define the given parameters:- Volume: V- Temperature: T- Energy range: [E, E + E]The partition function Z for a single particle in the canonical ensemble is given by:Z_1 = exp -E dwhere = 1/kT k is the Boltzmann constant , E is the energy of the particle, and d is the volume element in phase space.For an ideal gas, the energy is purely kinetic, so E = p^2/ 2m , where p is the momentum of the particle and m is its mass. The phase space volume element can be written as d = dxdydzdp_xdp_ydp_z, where x, y, and z are the spatial coordinates and p_x, p_y, and p_z are the momentum components.Now, let's calculate the partition function for a single particle:Z_1 = exp -p^2/ 2m dxdydzdp_xdp_ydp_zThis integral can be separated into spatial and momentum components:Z_1 = dxdydz * exp -p^2/ 2m dp_xdp_ydp_z The spatial integral is simply the volume V:dxdydz = VThe momentum integral is a Gaussian integral, which can be solved analytically: exp -p^2/ 2m dp_xdp_ydp_z = 2m/kT ^3/2 So, the partition function for a single particle is:Z_1 = V * 2m/kT ^3/2 For a system of two non-interacting particles, the partition function is the product of the partition functions of the individual particles:Z_2 = Z_1^2 = V^2 * 2m/kT ^3Now, let's calculate the probability of finding the system in the energy range [E, E + E]. The probability is given by the ratio of the number of states in the energy range to the total number of states, which can be expressed as:P E = number of states in [E, E + E] / total number of states Using the partition function, we can write this as:P E = Z_2 E / Z_2To find Z_2 E , we need to calculate the change in the partition function within the energy range [E, E + E]. Since the energy of the two-particle system is the sum of the energies of the individual particles, we can write:Z_2 E = Z_1 E1 * Z_1 E2 where E1 and E2 are the energies of the two particles, and Z_1 E1 and Z_1 E2 are the changes in the partition functions of the individual particles within the energy range [E1, E1 + E] and [E2, E2 + E], respectively.Now, we can calculate the probability:P E = Z_1 E1 * Z_1 E2 / Z_2This expression gives the probability of finding a system composed of two ideal gas particles in a given volume, temperature, and energy range using statistical mechanics and the canonical ensemble. The result depends on the specific values of the volume, temperature, and energy range, as well as the mass of the particles.