Both heat and work have intuitive definitions. However, we need to forego those as we study thermodynamics, because they can be misleading if not used carefully. To that end, we will rigorously define both concepts here.

Heat is the transfer of energy to a system via thermal contact with a reservoir. Work is the transfer of energy to a system via a change in the parameters of the system, such as volume.

This seemingly small distinction has significant consequences. Remember that a transfer of energy from a reservoir must obey the thermodynamic identity (taken for constant N and V), dU = τ dσ. Therefore a change in energy, i.e. a heat transfer, is accompanied by an entropy transfer. The addition of work, however, can't change the entropy of the system since we are only changing the external environment of the system.

We can look at the thermodynamic identity in a new way. The first term, τ dσ, can be thought of as the heat input, written dQ. The second term, - p dV, can be thought of as the work input, written dW. The third term, μ dN, can be thought of as the chemical work input, written dW_c. Therefore the total change in energy is due entirely to the sum of the heat inputted, work, and chemical work done on the system.

We need to think of entropy in a new way, though it is yet the same fundamentally as before. Entropy cannot build up indefinitely in a system. If it is introduced accompanying some heat input, it must eventually be released from the system.

This restriction does not affect the conversion of work into work, however. A plant that converts the rush of a river into electricity does not have to worry about entropy. Similarly, conversion of work into heat does not lead to a buildup of entropy. Conversion of heat to work, however, the basic process of a heat engine, must be done carefully to avoid buildup of entropy.

In fact, heat cannot be completely converted into work. Some heat must also be outputted as heat, to carry the entropy back out of the system. We can rewrite part of the thermodynamic identity as: σ_in = Q_in/τ_in. We want some of the input heat Q_in to be converted into work, so we know that Q_out will be less than Q_in.

We want all of the entropy to be extracted, however, and so we want σ_in = σ_out. The only way to accomplish such a feat is to have τ_in > τ_out. For this reason, we replace all of the "in" subscripts by "h", standing for "high temperature", and the "out" subscripts by "l", to indicate "low temperature".

The work that we actually get out in a heat engine is the difference between the input and output heat W = Q_h - Q_l = Q_h. Ideally, we would want W = Q_h, for in that case the system would be completely efficient.

For that reason, we define the Carnot efficiency, η_C, to be the ratio of the work to the input heat:

Some processes occur within an engine that create entropy irreversibly. Friction is a good example of such an unwanted source of entropy. We therefore can say that the actual efficiency of an engine is only as good or worse than the Carnot efficiency: η≤η_C. This relation is known as the Carnot Inequality.

Therefore a heat engine is a device that takes an input of heat at a high temperature, converts the heat partially to work, and expels heat at a lower temperature to maintain constant entropy inside the device. The lower temperature cannot practically be lower than that of the environment because the heat must eventually be dumped somewhere. Therefore the higher temperature is typically quite hot, usually many hundreds of Kelvin.

We can compress all of this information into a tidy diagram. Take some time to understand the diagram and what is represented there.

Looking back at Figure 1.1, there is nothing to prevent us from trying to reverse the process entirely. That is, we could try to input work to take heat from a low temperature to a high temperature. The only difference in the diagram is that the unwanted entropy created inside the device never works to our advantage and makes the outputted entropy greater than that inputted.

If you think about it, the description given above is exactly that of a refrigerator; it uses work (electrical work) to remove heat from inside the refrigerator, which is at a low temperature, and dump it into the environment at a higher temperature.

Two more common devices follow the same basic setup. An air conditioner is essentially a large refrigerator, where the inside of the refrigerator is exchanged for the inside of a house or car. A heat pump is an air conditioner in which we now swap the input and output. Therefore we extract heat from a cold environment to heat a warmer environment.

All of these devices work in essentially the same manner, which is a beautiful result of thermodynamics.