In this paper, we investigate a specific type of interacting particle system: coalescent processes. Specifically, we formally define this process using Markov chains and Poisson arrival processes, and provide an example of such a system, constructing the corresponding model in the process. To condense down the numerous aspects of physical particle motion and proximity, a rate function is utilized, leading to the Smoluchowski coagulation equations which determine the dynamics of the system. The paper also examines the theoretical aspects of coalescent processes by analyzing the state space size of the system and connecting it to the Bell numbers. We also formally define the Marcus-Lushnikov auxiliary process and explain its importance as a solution to inefficient modeling of the state space of the process. Following, the paper introduces the theoretical concept of the hydrodynamic limit and rigorously finds this limit for Kingman coalescence. Finally, the idea of gelation to explain large-scale particle system behavior is analyzed, including its relation to unique solutions of the Smoluchowski coagulation equations and its meaning in an applied sense.