- This study proposes a novel approach to quantifying uncertainties of constitutive relations inferred from noisy experimental data using inverse modelling. We focus on electrochemical systems in which charged species (e.g., Lithium ions) are transported in electrolyte solutions under an applied current. Such systems are typically described by the Planck-Nernst equation in which the unknown material properties are the diffusion coefficient and the transference number assumed constant or concentration-dependent. These material properties can be optimally reconstructed from time- and space-resolved concentration profiles measured during experiments using the Magnetic Resonance Imaging (MRI) technique. However, since the measurement data is usually noisy, it is important to quantify how the presence of noise affects the uncertainty of the reconstructed material properties. We address this problem by developing a state-of-the-art Bayesian approach to uncertainty quantification in which the reconstructed material properties are recast in terms of probability distributions, allowing us to rigorously determine suitable confidence intervals. The proposed approach is first thoroughly validated using "manufactured" data exhibiting the expected behavior as the magnitude of noise is varied. Then, this approach is applied to quantify the uncertainty of the diffusion coefficient and the transference number reconstructed from experimental data revealing interesting insights.