In this paper we investigate the case of IEEE 802.11-based WLANs and first show that, given the existing diversity of power consumption figures among mobile devices, performing a fair allocation of resources among devices is challenging. We then propose a criterion to objectively balance between the most energy-efficient configuration (where all resources are given to the single most energy efficient device) and the throughput-optimal allocation (where all devices evenly share the resources regardless of their power consumption). We derive a closed-form expression for the optimal configuration of the WLANs with respect to the energy-efficiency criterion. We validate our analysis through simulations, and show that our approach betters the prevalent allocation schemes discussed in literature in terms of energy efficiency, while maintaining the notion of fairness among competing devices.