The rare earth extraction process based on mechanism models does not consider mixer's energy efficiency in the extraction tank leading to a large error with the solvable content at each level and the industrial production data. This paper introduces the concept of energy efficiency separation coefficient, establishes a model which meets the actual content of rare earth elements in each extraction tank, and constructs an optimized objective function for solving the energy efficiency separation coefficient. Aiming at the multi-peak and multi-variable characteristics of the optimized objective function, an improved chaotic double mutation adaptive differential evolution algorithm(LCTADE) with linear population reduction is proposed. Firstly, the chaotic sequence is embedded in the improved algorithm to generate the initial population, which enhances the diversity of the population. Then in view of the different performance requirements of the mutation strategy in different evolution stages, a method based on the DE/current-to-pbest/1 and DE/rand/1 double mutation strategy is proposed. Finally, different parameter adaptive methods are used to set the $ F $, CR and NP values. In the simulation experiment, the function of the improved LCTADE on CEC2017 is tested and compared with other algorithms. The results show that the LCTADE is significantly better than other algorithms. The proposed algorithm is used to solve the energy efficiency separation coefficient of rare earth extraction process and optimize the objective functions, which shows its effectiveness.