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The article aimed to develop an accurate approximation of the fractional derivative with a non-singular kernel (the Rabotnov fractional-exponential formula), and show how to use it to solve numerically the blood ethanol concentration system. This model can be represented by a system of fractional differential equations. First, we created a formula for the fractional derivative of a polynomial function $ t^{p} $ using the Rabotnov exponential kernel. We used the shifted Vieta-Lucas polynomials as basis functions on the spectral collocation method in this work. By solving the specified model, this technique generates a system of algebraic equations. We evaluated the absolute and relative errors to estimate the accuracy and efficiency of the given procedure. The results point to the technique's potential as a tool for numerically treating these models.