Settle a (stupid, low stakes) bet. Question is in the title.
If you have a glass of water and leave it sitting around on your kitchen table will the temperature of the water eventually be the same as the ambient air temperature of the room it’s in?
What if it’s a larger container? A gallon jug, or 5 gallon pail?
If the temperature of the air is stable for the entire length of time (which will depend on the size of the glass/bucket, then I would say yes, given sufficient time.
That’s sure what happens when I leave a bottle of milk or wine out. 
And depending on the volume of water involved and the difference between the initial temperature of the water and the ambient temperature of the room, “sufficient time” may not be very long at all.
Never. If the temperature in the room stays constant, the water will forever be at a slightly lower temperature. The water temperature will approach room temperature asymtotically.
Also, trick question, trick answer, the water will change the air temperature, too!
If the water is warmer than the air, the water will cool down…but the air will heat up.
I’m not sure I agree with Try2BComprehensive that the answer is “never.” It’s only asymptotic to a “granular” degree. Eventually, the overall molecular motion will damp down. At least in classical terms.
In quantum terms, “never,” because there will always be some randomness that never wholly damps down. One pesky molecule will decide to go skittering off at some uncertain momentum.
To within any meaningful and measureable degree, yes. The two will get so close to the same temperature that no conceivable human agency could ever tell the difference.
If the relative humidity in the surrounding air is less than 100%, then there will be some evaporation from the water cooling the water, so it will be slightly less than the air temperature, until all the water has evaporated.
If there is evaporation, something gets colder, but is it the air or the water? Or both, in what ratio?
Conversely, if you have super-saturated air, water will/may condense out somewhere (possibly requires nucleation). Is the air warmer, or the condensate? Or both?
You sir underestimate the Justice League!
Evaporative cooling will keep the water below the air temp unless it’s at or close to 100% RH.
If the water is warmer than the air to start with it will eventually end up colder than the air due to evaporative cooling. That means at some point the air and the water will be the same temperature.
The water. Those water molecules that are hotter, and thus faster, will escape more often than the colder ones and thus lower the average temperature of the water.
I think Tripolar nailed it!
Very good answers - thank you.
As a follow up, how much cooler would the water be, generally. Assume the glass (or whatever) is in a room at 72 degrees, 30% relative humidity (or whatever is standard in a house). A couple of degrees? Less than a degree? 5 degrees?
The way that I think of it is like this: if you have two schools (the liquid water and the air), and you take out any student from the first school who has an A+ average and send him to the second school, will the first school’s average grade go down? Yes, it will.
That depends on the water’s initial temperature, how long it’s been out, and the humidity of the air. Any answer at all is possible.
The original question assumed we waited until “eventually”.
“Temperature” is an average property of a bulk material, not one of individual molecules. So we have to be careful to ensure we here, and also the OP & his betting counterparty agree on what we’re measuring and at what level of detail. As **Trinopus **said above ref quantum stuff, if you look closely enough at reality, any reality, you’ll find that nothing is ever stable, constant, or equal to anything else[sup]1[/sup].
My take is that the evaporative cooling difference cited by **Giles **and kanicbird, while 100% real, is very, very small. For any plausible room-like initial conditions, including those specified by the quote above, evaporative cooling of a still glass of water will make that water a tiny, almost undetectable fraction of a degree lower than the air temp.
In fact, at the level of detail and precision where we can detect a difference in temperature between air and water, we also have to start talking about the many different temperatures in different parts of the water & in different areas within the room.
IMO it’s pretty clear the OP is thinking about gross differences, the kinds one could detect with a fever thermometer or by sticking your finger in the water. At that level of detail, whether the water started at 0.1C or 99.9C within a couple hours the air and water will reach the same measured temperature. Then the water will slowly evaporate, remaining at the same measured temperature as the air until the water is gone.
Thank you very much for this detailed explanation. FWIW, I won the bet! 
It can also depend on the amount of exposed surface area and the elevation. 
My vote for most normal glasses on normal Earth conditions is less than a degree, though. Evaporative cooling systems never give you better than a 20-degree difference even in low humidity. A glass of water isn’t designed for evaporation, though. It’ll take in heat from the air many times faster than it loses heat through evaporation.
In
:: spits in contempt ::
Fricking Aquaman.