This section defines heat, “the spontaneous flow of energy from one object to another caused by the difference in temperature between the objects,” and work, any non-heat transfer of energy into or out of a system. It also introduces the first law of thermodynamics, $\Delta U = Q + W$, where U is the total energy of the system, Q is the energy that enters the system as heat, and W is the energy that enters the system as work. Schroeder points out that these definitions of heat and work are counter to the way we use the terms in common speech. For example, if you rub your cold hands together to warm them up, by these definitions there is no heating being done to the system, because the change in temperature isn’t due to spontaneous energy flow. Rather, your hands are being warmed by work.

Schroeder goes on to introduce the units Joule, calorie (equal to 4.186 J), and kilocalorie (also known as the food calorie, 4186 J); and the three types of heat transfer, conduction (transfer by molecular contact), convection (transfer by the bulk motion of a gas or liquid), and radiation (emission of electromagnetic waves.)

Problem 1.26: A battery is connected in series to a resistor, which is immersed in water (to prepare a hot cup of tea). Would you classify the flow of energy from the batter to the resistor as “heat” or “work”? What about the flow of energy from the resistor to the water?

The flow of energy from the battery to the resistor is work, because the energy transferred by the electrons moving through the resistor is driven by the voltage difference, it doesn’t occur spontaneously due to a temperature difference. The flow of energy from the resistors to the water, though, is heat.

Problem 1.27: Give an example of a process in which no heat is added to a system, but its temperature increases. Then give an example of the opposite: a process in which heat is added to a system but its temperature does not change.

Imagine you have a cylinder of compressed air that is at room temperature. If you open the valve and start letting air out, the pressure within the cylinder will drop. By the ideal gas law, the temperature of the air inside the cylinder will drop proportionally with the pressure. (This is why the propane tanks attached to gas grills occasionally ice up while the grill is in use.) The opposite condition would hold if you pumped air from the room into the cylinder. Then the temperature of the gas in the cylinder would rise proportionally with the pressure, but no heat would be added.

A system where heat is added but the temperature doesn’t change would be one where either the heat energy is being compensated for by work, or where the heat energy is causing a non-temperature change, like a phase change. So for example if you boil water, energy is constantly flowing into the water due to heat, but the temperature stays at 100°C. The energy goes into changing the water from liquid phase to gas phase.

Problem 1.28: Estimate how long it should take to bring a cup of water to boiling temperature in a typical 600-watt microwave oven, assuming that all the energy ends up in the water. (Assume any reasonable initial temperature for the water.) Explain why no heat is involved in this process.

A calorie is the amount of energy needed to raise one gram of water by 1° C. A watt is one joule per second. One cup of water is 236.6 ml, which is also 236.6 grams. If we assume the water is at 21° C, then the the amount of energy needed to bring the water to boiling is 236.6 x 79 = 18691.4 calories, which is 78,242 joule. Dividing by 600, we get 130 seconds to heat the water to boiling. No heat is involved because the microwaves that are physically vibrating the polar water molecules are being generated, not emitted spontaneously. This is work, just like when you rub your hands together so that the friction warms them. This isn’t a spontaneous transfer of energy due to temperature, it’s a driven energy transfer.

Problem 1.29: A cup containing 200 g of water is sitting on your dining room table. After carefully measuring its temperature to be 20° C, you leave the room. Returning ten minutes later, you measure the temperature again and find that it is now 25° C. What can you conclude about the amount of heat added to the water? (Hint: This is a trick question)

You can infer that the net heat added to the water was less than or equal to 1000 calories. It’s possible that no heat was added, and the temperature increase was due entirely to work. It’s possible that some work was done, and also some heat added. It’s possible that more than 1000 calories of heat were added, and then removed again. You can’t know for sure anything other than that there are 1000 new calories of energy, some or all or none of which could have come from heat.