Open-domain chatbots engage with users in natural conversations to socialize and establish bonds. However, designing and developing an effective open-domain chatbot is challenging. It is unclear what qualities of a chatbot most correspond to users' expectations and preferences. Even though existing work has considered a wide range of aspects, some key components are still missing. For example, the role of chatbots' ability to communicate with humans at the emotional level remains an open subject of study. Furthermore, these trait qualities are likely to cover several dimensions. It is crucial to understand how the different qualities relate and interact with each other and what the core aspects would be. For this purpose, we first designed an exploratory user study aimed at gaining a basic understanding of the desired qualities of chatbots with a special focus on their emotional intelligence. Using the findings from the first study, we constructed a model of the desired traits by carefully selecting a set of features. With the help of a large-scale survey and structural equation modeling, we further validated the model using data collected from the survey. The final outcome is called the PEACE model (Politeness, Entertainment, Attentive Curiosity, and Empathy). By analyzing the dependencies between the different PEACE constructs, we shed light on the importance of and interplay between the chatbots' qualities and the effect of users' attitudes and concerns on their expectations of the technology. Not only PEACE defines the key ingredients of the social qualities of a chatbot, it also helped us derive a set of design implications useful for the development of socially adequate and emotionally aware open-domain chatbots.