Behold, our secret weapon in the furnace wars. (This photo will look familiar to those who read Blog Action Day: Walking My Watershed. I hope you read it; it's one of the most popular posts on this blog.)
Our bathroom sink and one of the twin sinks in our kitchen hold about 10 liters of water. Suppose we fill it with the tap water at 50C (122F). That's 10 kg of water with a heat capacity of 4.2 kJ/(kg-K). One liter of water will release 4.2 kiloJoules of energy for every degree C it cools. Suppose the water cools to a comfortable 25C (77F), then it releases
10 liters * 25 degrees * 4.2 kJ/(kg-K) = 1050 kJoules.
Our kitchen is roughly 12x12 feet with 8 feet ceilings. Convert that into meters and we get an air volume of 32621 liters. The molecular mass of air is about 0.029 kg/mole. (Air is a mixture of gases so this is the weighted mass of it's constituent gases.)
One mole of gas at standard temperature and pressure (STP), a comfortable temperature for humans at sea level pressure, fills up 22.14 liters. So the air in our kitchen weighs
32621 liters * 0.029 kg/mole / 22.14 liters/mole = 42.7 kg.
The heat capacity of air at STP is 1.00 kj/(kg-K), so the 1050 kiloJoules of energy released by the cooling sink water can heat the air in the kitchen
1050 kJ / (42.7 kg * 1.00 kJ/(kg-K)) = 24.6 C
If the air in the kitchen started at 20 C, and is warmed another 24.6 C, then it would be a toasty 112 F.
The kitchen does not get that warm because some of that heat is transferred to the porcelain sink, heat capacity 1.07 kJ/(kg-K), furnishings, etc. But, you get the general idea. You used an awful lot of energy to heat that hot water, so you might as well get as much of that energy back before you send it down the drain.
Fancy new houses might have an expensive heat exchanger system*, using waste heat from the water leaving the house to heat up the water coming into the house. But, we have an older and simpler house; we use the same cheap and effective technology our grandparents used, our brains.
Now calculate the equilibrium temperature when 10 kg of water at 50 C meets 42 kg of air at 20 C. Assume that it is a closed system (no heat loss to the sink or furnishings). Leave your answer in the comments.
* Actually, heat exchanger systems are not that high tech or new. They just run tubes of hot and cold water around each other. I first read about that in the 1970s. But, they did not gain traction in mass-market housing. They were recently resurrected in expensive and huge LEED homes.
Alison makes a good point in her comment. You do get much bang for your energy buck if you let the bathwater cool before you send it down the drain. When we used to take baths, that's what we did in the winter time. We take showers now--with a very low flow shower head. If you don't mind rinsing the soap off your feet in the faucet before you exit the tub, you can recoup a great deal of heat from your shower water, too.