Nearly all studies of and available data for pressure-dependent reactions focus on pure bath gases. Of the comparatively fewer studies on bath gas mixtures, important to combustion and planetary atmospheres, nearly all focus on single-channel reactions. The present study explores, and seeks reliable representations of, bath gas mixture effects on multichannel reactions. Analytical and numerical solutions of the master equation here reveal several unique manifestations of mixture effects for multichannel reactions, including behavior completely opposite to trends observed for single-channel reactions. The most common way of evaluating mixture rate constants from data for pure components, the classic linear mixture rule, is found to yield errors exceeding a factor of similar to 10. A new linear mixture rule based on the reduced pressure, instead of the absolute pressure, is found to be accurate within similar to 30% for rate constants (and similar to 50% for the branching ratio). A new nonlinear mixture rule that additionally incorporates analytically derived activity coefficients is found to be accurate within similar to 10% for rate constants and branching ratios. These new mixture rules are therefore recommended for use in fundamental and applied chemical kinetics investigations of reacting mixtures, including reacting flow codes and experimental interpretations of third-body efficiencies.