In this project, the time-dependent one-dimensional heat equation with internal heating is solved using eigenfunction expansion, according to the thermoelectric boundary conditions. This derivation of the equation describing time-dependent heat flow in a thermoelectric sample or device yields a framework that scientists can use (by entering their own parameters into the equations) to predict the behavior of a system or to verify numerical calculations. Allowing scientists to predict the behavior of a system can help in decision making over whether a particular experiment is worthy of the time to construct and execute it. For experimentalists, it is valuable as a tool for comparison to validate the results of an experiment. The calculations done in this derivation can be applied to pulsed cooling systems, the analysis of Z-meter measurements, and other transient techniques that have yet to be invented. The vast majority of the calculations in this derivation were done by hand, but the parts that required numerical solutions, plotting, or powerful computation, were done using Mathematica 8. The process of filling in all the steps needed to arrive at a solution to the time-dependent heat equation for thermoelectrics yields many insights to the behavior of the various components of the system and provides a deeper understanding of such systems in general.