Usually not.
You are required to do all you can to mitigate the damages, meaning you do your best to rerent the unit. Once you find another tenant who starts paying rent, the person who broke the lease is no longer responsible. You aren't allowed to collect what would amount to be rent from two different tenants at once.
However, you also can charge the person who broke the lease reasonable expenses you had for rerenting the unit, such as advertising or a fee paid to a property manager who found the new tenant.
Some persons have tried to collect utilities. That is a fairly iffy thing.
Keep in mind that many things are not specifically addressed in Missouri law - so much of what the decision is depends on the judge, and how that particular judge is feeling that day.
Also, in order to sue for broken lease damages, you need to know where the departed tenant is - since service is required. You also need someone who is collectible - and many if not most of those who break a lease aren't.
So, you have to decide if it is worth your money and effort to try to do this.
In almost 30 years of landlording, I have only bothered to do this once.
--205.188.xxx.xx