The irreversible process of mixing two gases at different temperatures can be described using non-equilibrium thermodynamics by considering the transport phenomena and entropy production during the mixing process. In this case, we will focus on heat conduction and diffusion, as these are the primary mechanisms responsible for the mixing of gases.Let's consider two ideal gases A and B, initially separated by a partition in a container. Gas A has an initial temperature T_A, pressure P_A, and volume V_A, while gas B has an initial temperature T_B, pressure P_B, and volume V_B. When the partition is removed, the gases will mix irreversibly due to the temperature and concentration gradients.Quantitative analysis:1. Heat conduction: The temperature gradient between the two gases drives heat conduction. The heat conduction can be described by Fourier's law:q = -k * dT/dxwhere q is the heat flux, k is the thermal conductivity, and dT/dx is the temperature gradient.2. Diffusion: The concentration gradient between the two gases drives diffusion. Fick's law describes the diffusion process:J = -D * dC/dxwhere J is the diffusion flux, D is the diffusion coefficient, and dC/dx is the concentration gradient.During the mixing process, the total entropy change S can be divided into two parts: the entropy change due to heat conduction S_heat and the entropy change due to diffusion S_diff .S = S_heat + S_diffFor an ideal gas, the entropy change due to heat conduction can be calculated using the following equation:S_heat = n * C_v * ln T_final/T_initial where n is the number of moles, C_v is the heat capacity at constant volume, T_final is the final temperature, and T_initial is the initial temperature.The entropy change due to diffusion can be calculated using the following equation:S_diff = -n * R * ln x_final/x_initial where R is the gas constant, and x_final and x_initial are the final and initial mole fractions, respectively.Consequences of irreversible mixing:1. Entropy production: The irreversible mixing of gases leads to an increase in the total entropy of the system. This is in accordance with the second law of thermodynamics, which states that the entropy of an isolated system always increases during an irreversible process.2. Temperature and pressure equilibration: As the gases mix, their temperatures and pressures will equilibrate to a common final temperature T_final and pressure P_final . This is due to the heat conduction and diffusion processes that occur during mixing.3. Loss of available work: The irreversible mixing of gases results in a loss of available work that could have been extracted from the system if the process were reversible. This is because the entropy increase during the irreversible process is associated with a decrease in the Gibbs free energy, which represents the maximum work that can be extracted from a system.