The effect of temperature on the rate of polymerization of vinyl monomers using a free radical mechanism can be understood by examining the various steps involved in the process. The free radical polymerization mechanism consists of three main steps: initiation, propagation, and termination.1. Initiation: In this step, a free radical initiator such as a peroxide or an azo compound decomposes into two free radicals upon heating. These free radicals can then react with the vinyl monomer to form a new radical species.2. Propagation: The newly formed radical species reacts with another vinyl monomer, forming a covalent bond and generating a new radical species. This process continues, leading to the growth of the polymer chain.3. Termination: The polymerization process ends when two radical species react with each other, forming a covalent bond and terminating the chain growth.The effect of temperature on the rate of polymerization can be analyzed by considering the following factors:A. Activation energy: The initiation step requires a certain amount of energy to break the initiator's chemical bonds and generate free radicals. As the temperature increases, the kinetic energy of the molecules also increases, making it easier for the initiator to decompose and generate free radicals. This leads to a higher rate of initiation and, consequently, a higher rate of polymerization.B. Reaction rate constants: The rate constants for the propagation and termination steps are also temperature-dependent. According to the Arrhenius equation, the rate constant k is related to temperature T by the following equation:k = Ae^-Ea/RT where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. As the temperature increases, the rate constants for both propagation and termination increase, leading to a higher rate of polymerization.C. Chain length and viscosity: As the temperature increases, the average chain length of the polymer decreases due to the increased rate of termination. This results in a decrease in the viscosity of the reaction mixture, which can further enhance the rate of polymerization by facilitating the diffusion of reactants.Graphical representation:A typical graph representing the effect of temperature on the rate of polymerization would show the rate of polymerization Rp on the y-axis and the temperature T on the x-axis. The graph would generally exhibit an upward trend, indicating that the rate of polymerization increases with increasing temperature. However, the relationship may not be linear, as the rate constants for initiation, propagation, and termination steps may have different temperature dependencies.In conclusion, the rate of polymerization of vinyl monomers using a free radical mechanism is significantly affected by temperature. An increase in temperature generally leads to a higher rate of polymerization due to the increased rate of initiation, propagation, and termination steps, as well as the reduced viscosity of the reaction mixture.