Is there an event for when a Chrome Extension popup is closed?

It's as simple as that:

// popup.js
chrome.runtime.connect({ name: "popup" });

// background.js
chrome.runtime.onConnect.addListener(function(port) {
    if (port.name === "popup") {
        port.onDisconnect.addListener(function() {
           console.log("popup has been closed")
        });
    }
});

Note that there are some edge cases where this method doesn't work. E.g. when having the popup open and then switching tabs by using a keyboard shortcut such as ctrl + t for example.

You can try this. Connect to your background page with chrome.runtime.connect (or chrome.extension.connect before Chrome 26) and port.onDisconnect will be fired in your background page when the popup is closed.