∫ Heating a Room

How do we heat a room? If your answer is “by adding energy to the room,” then there is good news and bad news. The good news is you are thinking about the question like most physicists and engineers. The bad news is you’re wrong.

In a recent article in the American Journal of Physics, physicists Hans Kreuzer and Stephen Payne revisited this simple thermodynamics question that most scientists have been answering incorrectly for at least 70 years.

The confusion is rooted in the difference between two concepts: energy and temperature. The answer to the question: “How do we heat a room” is in fact, “by increasing the temperature.” But it is not “by increasing the internal energy of the room.” How is this possible?

The answer can be seen in the ideal gas equation, an equation that surfaces in hundreds of energy optimization problems and is an excellent approximation for the behavior of air at room temperature. The equation is:

PV = nRT

Where:

  • P is the pressure
  • V the volume
  • n the number of gas particles (as measured in moles)
  • R is a constant
  • T is the temperature

Notice that both sides of the equation are in units of energy. If you have a pressure throughout a container of fixed volume, then the energy inside that container is simply the pressure times the volume.

Now let’s consider a living room. The volume of a room is essentially constant: the walls, floor, and ceiling are not moving much. At least we hope not. Additionally, the pressure in the room cannot be greatly different than it is outside, or else the windows would shatter. An increase in pressure in the room will result in air escaping through cracks under doors and windows in order to equalize the pressure. So both pressure and volume are staying constant. Which means the energy inside the room stays constant.

Let’s look at the other half of the equation, nRT. This is also a measure of energy. And because it is equal to PV, nRT must also not change inside a room. R is just a number so it stays fixed. But T must be going up because we’re feeling warmer and watching a thermometer rise. So if PV is constant, R is fixed, and T is increasing, then n must be decreasing – or else the equation will not hold.

This means that when we heat a room, the number of particles in the room is going down, while the energy of the room stays fixed. The remaining particles each have more energy than they did before the heater turned on, but there are fewer of them. Air is expelled outside. In fact, the energy used by the heater to increase the temperature of your living room, actually ends up outdoors, in the form of more particles (a larger n) bouncing around.

Pretty strange, no? What’s going on behind the thermodynamic scenes is a change in entropy. The entropy in the room decreases as the temperature goes up. Doesn’t the second law of thermodynamics teach that entropy always increases? Only in closed systems. In the case of heating a room, the entropy of the room decreases, but the overall entropy of the room and the outside (the system) rises.

What about the heat needed to heat the walls themselves? Kreuzer and Payne, as well as several thermodynamics textbooks, treat this interesting question in greater detail. They conclude that for a typical room, about 10% of the energy needed to raise the temperature of the air in the room is needed to raise the temperature of the walls. Additionally, they put in perspective heat lost through windows, cracks, and doors. Over the course of a day, heat lost through conduction and convection through these openings can exceed the energy needed to increase the temperature of the room. So make sure to insulate your home and put weather stripping on doors and windows!

Thermodynamics may make it impossible to increase the energy of your living room, but an efficient furnace and thorough insulation will minimize the energy rushing into the great outdoors.

6 Responses to ∫ Heating a Room

  1. Super interesting explanation of something I never quite wrapped my brain around until now.

    At some point you guys should slap all your articles together and publish a textbook!

    Reply
    • brenden says:

      Thanks for the encouragement Parker! We are currently planning a series of video lectures, so stay tuned for another couple months.

      Reply
  2. Ted Hesser says:

    Fascinating article. One of substance, insight and intrigue. Leagues above the noisy chatter of internet blogs :)

    I find the negative entropy idea to be foundational to our experiences on earth. Earth is actually a giant pocket of negative entropy. We, human beings, are also pockets of negative entropy. It takes a whole lot of cellular organization for us to exist, and not of the form that Verizon might promote. Everything that we love and like in life, yummy food, ipods, warm rooms, cold beers, hot showers, Amory Lovin’s bannana greenhouse room. All of these things exist in negative entropy states, formed at the expense of increased entropy, that is eventually expelled into the cosmos.

    I wonder if there is a way to leverage this insight to build a better mousetrap, i.e. optimize a room-heating device around decreasing entropy, as opposed to expelling heat. food for thought.

    Reply
    • raphael says:

      Hi Ted. Very interesting thought. It sounds like you’re a true believer in Maxwell’s Demon! I’m afraid trying to outsmart entropy rarely turns out well and never does so in an insolated system. The room isn’t a closed system, but as long as P and V are fixed, no matter what you do, your energy is going outside.

      Reply

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