Now that we have seen how to calculate internal energy, heat, and work done for a thermodynamic system undergoing change during some process, we can see how these quantities interact to affect the amount of change that can occur. This interaction is given by the first law of thermodynamics. British scientist and novelist C. P. Snow (1905–1980) is credited with a joke about the four laws of thermodynamics. His humorous statement of the first law of thermodynamics is stated “you can’t win,” or in other words, you cannot get more energy out of a system than you put into it. We will see in this chapter how internal energy, heat, and work all play a role in the first law of thermodynamics.
Suppose Q represents the heat exchanged between a system and the environment, and W is the work done by or on the system. The first law states that the change in internal energy of that system is given by Q-W. Since added heat increases the internal energy of a system, Q is positive when it is added to the system and negative when it is removed from the system.
When a gas expands, it does work and its internal energy decreases. Thus, W is positive when work is done by the system and negative when work is done on the system. This sign convention is summarized in (Figure). The first law of thermodynamics is stated as follows:
First Law of Thermodynamics
Associated with every equilibrium state of a system is its internal energy {E}_{\text{int}}. The change in {E}_{\text{int}} for any transition between two equilibrium states is
\text{Δ}{E}_{\text{int}}=Q-W
where Q and W represent, respectively, the heat exchanged by the system and the work done by or on the system.
Thermodynamic Sign Conventions for Heat and Work
Process Convention
Heat added to system Q>0
Heat removed from system Q<0
Work done by system W>0
Work done on system W<0
The first law is a statement of energy conservation. It tells us that a system can exchange energy with its surroundings by the transmission of heat and by the performance of work. The net energy exchanged is then equal to the change in the total mechanical energy of the molecules of the system (i.e., the system’s internal energy). Thus, if a system is isolated, its internal energy must remain constant.
Although Q and W both depend on the thermodynamic path taken between two equilibrium states, their difference Q-W does not. (Figure) shows the pV diagram of a system that is making the transition from A to B repeatedly along different thermodynamic paths. Along path 1, the system absorbs heat {Q}_{1} and does work {W}_{1}; along path 2, it absorbs heat {Q}_{2} and does work {W}_{2}, and so on. The values of {Q}_{i} and {W}_{i} may vary from path to path, but we have
{Q}_{1}-{W}_{1}={Q}_{2}-{W}_{2}=\text{⋯}={Q}_{i}-{W}_{i}=\text{⋯}\text{,}
or
\text{Δ}{E}_{\text{int}1}=\text{Δ}{E}_{\text{int}2}=\text{⋯}=\text{Δ}{E}_{\text{int}i}=\text{⋯}\text{.}
That is, the change in the internal energy of the system between A and B is path independent. In the chapter on potential energy and the conservation of energy, we encountered another path-independent quantity: the change in potential energy between two arbitrary points in space. This change represents the negative of the work done by a conservative force between the two points. The potential energy is a function of spatial coordinates, whereas the internal energy is a function of thermodynamic variables. For example, we might write {E}_{\text{int}}\left(T,p\right) for the internal energy. Functions such as internal energy and potential energy are known as state functions because their values depend solely on the state of the system.
Different thermodynamic paths taken by a system in going from state A to state B. For all transitions, the change in the internal energy of the system \text{Δ}{E}_{\text{int}}=Q-W is the same.
The figure is a graph of p on the vertical as a function of V on the horizontal axis. Six different curves are shown, all connecting a point A on the graph to a point B. The pressure at A is larger than at B, and the volume at A is lower than at B. Curve 1 goes up and curves around to reach B from above. Curve 2 is similar to 1 but not as curved. Curve 3 is a straight line from A to B. Curve 4 wiggles a bit below the straight line of curve 3. Curve 5 bends down and around to B, reaching it from below. Curve 6 is similar to curve 5 but goes farther out.
Often the first law is used in its differential form, which is
d{E}_{\text{int}}=dQ-dW.
Here d{E}_{\text{int}} is an infinitesimal change in internal energy when an infinitesimal amount of heat dQ is exchanged with the system and an infinitesimal amount of work dW is done by (positive in sign) or on (negative in sign) the system.
