The ideal Bose-gas consists of massless particles with the law of dispersion omega (k)=Acdot k^2 , where is the particle wave vector and A is a given constant.Determine the heat capacity of the gas at a temperature T. Consider the spin of the particles to be equal to 0. Be careful:the assessment will take into account the progress of the solution;writing only the answer is not enough. please see the solution sheets in detail. | square File uploading will be available after the completion of the testing of the Second Stage Please enter at least 1 character in the text field and save the answer.

To determine the heat capacity of the ideal Bose-gas at temperature T, we need to consider the energy distribution of the particles in the gas.The energy of a particle in the ideal Bose-gas is given by the law of dispersion , where k is the wave vector of the particle and A is a given constant.The total energy of the gas can be calculated by summing up the energies of all the particles in the gas. The total energy is given by: where N is the total number of particles in the gas and is the wave vector of the i-th particle.The heat capacity of the gas is defined as the change in total energy with respect to temperature. Mathematically, it is given by: To calculate the heat capacity, we need to find the derivative of the total energy with respect to temperature. However, since the energy of each particle depends on the wave vector, which in turn depends on the temperature, we need to consider the distribution of wave vectors at temperature T.The distribution of wave vectors in the ideal Bose-gas is given by the Bose-Einstein distribution function: where is the inverse temperature and is the Boltzmann constant.Substituting the law of dispersion into the distribution function, we have: To calculate the total energy, we need to sum up the energies of all the particles in the gas, taking into account their distribution. Mathematically, it is given by: Since the number of particles N is very large, we can consider the sum as an integral over all possible wave vectors: Substituting the law of dispersion and the distribution function, we have: To calculate the heat capacity, we need to differentiate the total energy with respect to temperature: Differentiating the integral expression with respect to temperature, we get: Taking the partial derivative of the energy function with respect to temperature, we get: Simplifying the expression, we get: Substituting this expression into the integral for the heat capacity, we get: Evaluating the integral, we get: This integral can be evaluated numerically or using approximation methods. The final result will depend on the value of the constant A and the temperature T.