"How does distillation separate components in a mixture based on their boiling points? Explain the principle behind vapor$-$liquid equilibrium and its role in the distillation process. Discuss the factors influencing separation efficiency in distillation columns. Furthermore, elucidate the concept of mass transfer in chemical engineering and its importance in various unit operations. Provide an example of an industrial application where distillation plays a crucial role in separating components for product purification."