This study investigates the application of Physics-Informed Neural Networks (PINNs) to solve the differential algebraic equations (DAEs) governing the complex dynamics of a Two-Phase Reactor and Condenser with Recycling system (TPRCR system). The TPRCR system is characterized by highly nonlinear, stiff, and interdependent equations that describe variables such as concentrations, temperatures, and pressures. Traditional numerical methods often struggle with these equations due to their complexity, sensitivity, and potential partial unknowns, such as difficult-to-model transport phenomena and reaction kinetics. PINNs offer a unique data-driven approach by directly embedding the physical laws governing the TPRCR system into the neural network's training process, thereby bypassing the need for a fully explicit derivation from the governing equations. This makes them particularly effective for systems like TPRCR, where exact analytical solutions are elusive and traditional methods require significant computational effort. In this study, we trained multiple PINN models using the AdamW optimizer with a learning rate of 0.001 over 500 epochs. These models accurately predicted system dynamics, including concentrations, temperatures, and pressures. The root-mean-squared error was employed as the loss function to guide optimization and ensure high accuracy in predictions. Our results demonstrate that PINNs successfully capture the complex behaviour of the TPRCR system, with predictions aligning closely with those obtained from conventional DAE solutions. The computational efficiency and fast convergence of PINNs further highlight their potential for robust performance in chemical process modelling. This study underscores the value of PINNs in addressing the specific challenges posed by the TPRCR system, making them a promising tool for future research and industrial applications in chemical engineering.