We proposed a method to calculate the pressure, opening force, and leakage rate in a mechanical seal under the mixed lubricant state of a gas and liquid for the mechanical seal in a reactor cooling pump (RCP) during external power loss. We calculated the pressure by solving the nonlinear finite element equation composed of the linear Reynolds equation of an incompressible liquid lubricant and the nonlinear Reynolds equation of a compressible gas lubricant using the Newton–Raphson method. In addition, we calculated the temperature distribution by solving the two-dimensional energy equation utilizing the finite element method. Additionally, we included the turbulence effect in the incompressible liquid lubricant and the turbulence and slip effects in the compressible gas lubricant. The accuracy of the developed program was validated by comparing the simulated opening force and leakage rate of both the mechanical seal with the liquid lubricant and the mechanical seal with the gas lubricant with prior research. Our analysis shows that in high-temperature environments, the increase in the gas region at the lubrication surface leads to a decrease in pressure and opening force and an increase in the leakage rate. Conversely, as the outer pressure increases, the gas region decreases, resulting in an increase in pressure, opening force, and leakage rate.