The transportation sector accounts for 23% of global energy-related CO2 emissions, with road vehicles contributing 70\% of these emissions. Traffic Assignment (TA) models are essential for optimizing traffic flow, reducing congestion, and consequently lowering emissions from this sector. This study proposes a traffic management strategy aimed at reducing carbon emissions through speed limit controls in multi-modal freeway networks. We propose a method that solves the network User Equilibrium (UE) problem while optimizing carbon emissions by adjusting speed limits. By focusing on multi-modal traffic management and calculating carbon abatement curves, this approach provides practical tools for traffic authorities to design effective policies. The proposed methodology is applied to a realistic case study of a freeway network in Tilburg, The Netherlands. Numerical experiments are conducted under various scenarios, analyzing the impact of different objective functions and constraints on traffic flow, emissions, and travel time. The results show that incorporating speed controls and allowing flexibility in route choices through Bounded Rationality User Equilibrium (BRUE) significantly reduces emissions and total travel costs compared to traditional UE models. Specifically, the proposed method can reduce overall emissions by up to 19.5\% compared to the most optimized traditional UE models. These findings offer valuable insights for policymakers aiming to create eco-friendly and economically viable traffic management strategies.