You have just seen that when a system changes from a given initial state to a given final state, both the work W and the heat Q depend on the nature of the process. Experimentally, however, we find a surprising thing. The quantity Q − W is the same for all processes. It depends only on the initial and final states and does not depend at all on how the system gets from one to the other. All other combinations of Q and W, including Q alone, W alone, Q + W, and Q − 2W, are path dependent; only the quantity Q − W is not.
The quantity Q − W must represent a change in some intrinsic property of the system. We call this property the internal energy Eint and we write
Equation 18-26 is the first law of thermodynamics. If the thermodynamic system undergoes only a differential change, we can write the first law as*
The internal energy Eint of a system tends to increase if energy is added as heat Q and tends to decrease if energy is lost as work W done by the system.
In Lesson 8, we discussed the principle of energy conservation as it applies to isolated systems—that is, to systems in which no energy enters or leaves the system. The first law of thermodynamics is an extension of that principle to systems that are not isolated. In such cases, energy may be transferred into or out of the system as either work W or heat Q. In our statement of the first law of thermodynamics above, we assume that there are no changes in the kinetic energy or the potential energy of the system as a whole; that is, ΔK = ΔU = 0.
Before this lesson, the term work and the symbol W always meant the work done on a system. However, starting with Eq. 18-24 and continuing through the next two lessons about thermodynamics, we focus on the work done by a system, such as the gas in Fig. 18-13.
*Here dQ and dW, unlike dEint, are not true differentials; that is, there are no such functions as Q(p, V) and W(p, V) that depend only on the state of the system. The quantities dQ and dW are called inexact differentials and are usually represented by the symbols 
Q and W. For our purposes, we can treat them simply as infinitesimally small energy transfers.
The work done on a system is always the negative of the work done by the system, so if we rewrite Eq. 18-26 in terms of the work Won done on the system, we have ΔEint = Q + Won. This tells us the following: The internal energy of a system tends to increase if heat is absorbed by the system or if positive work is done on the system. Conversely, the internal energy tends to decrease if heat is lost by the system or if negative work is done on the system.
CHECKPOINT 5 The figure here shows four paths on a p–V diagram along which a gas can be taken from state i to state f. Rank the paths according to (a) the change ΔEint in the internal energy of the gas, (b) the work W done by the gas, and (c) the magnitude of the energy transferred as heat Q between the gas and its environment, greatest first.
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