Changes in State Functions

Although the changes in the two state functions are equal (and zero) in this example, this is not always the case. 

Because constant-pressure processes are so common (almost any process exposed to the atmosphere can be considered constant-pressure), the equality of q and AH is common. Because of this, it is also common to use the words endothermic and exothermic to describe processes for which the AH is positive or negative, respectively. However, this is only strictly correct for constant-pressure processes. 

Although we stated that we can know only the change in internal energy or enthalpy, so far we have mostly dealt with the overall change of a complete process. We have not considered infinitesimal changes in H or (/ in much detail. 

Which two do we pick for internal energy and enthalpy Although we can pick any two, in the mathematics that follow there will be advantages to picking a certain pair for each state function. For internal energy, we will use temperature and volume. For enthalpy, we will use temperature and pressure. 

The total differential of a state function is written as the sum of the derivative of the function with respect to each of its variables. For example, dU is equal to the change in U with respect to temperature at constant volume plus the change in U with respect to volume at constant temperature. For the change in U written as U T, V) — U T + dT, V + dV), the infinitesimal change in internal energy is 

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State functions (Frame 1, section 1.5) are amenable to the construction of thermochemical cycles since provided that we can devise alternative routes between the same initial and final states, then the changes in state functions via these two alternative routes will be identical. 

The equations developed in this section have been derived for mechanically reversible nonflow processes involving ideal gases. However, those equations which relate state functions only are valid for ideal gases regardless of the process and apply equally to reversible and irreversible flow and nonflow processes, because changes in state functions depend only on the initial and Anal states of the system. On the other hand, an equation for Q or W is specific to the case considered in its derivation. 

If tlie volume varies during the process but returns at the end of the process to its initial value, the process cannot rightly be called one of constant volume, even tliough Vi = Vi and A y = 0. However, changes in state functions or properties are independent of path, and are tlie same for all processes which result in the same change of state. Property changes are tlierefore... 

The enthalpy of reaction, AFf, is evaluated by the expression A// = E, which is analogous to the expression for the free energy of re tion, AG = E PiPj. Expressions for AK and AS are similar AV = E, Vi Vi and AS = E, Vi Si- Using these expressions, we can readily calculate the changes in state functions—AG, AH, AS, and AK—needed to find the state of a chemical reaction and the influences of temperature and pressure on that reaction. 

This section demonstrates calculations of changes in macroscopic properties caused during several specific reversible processes in ideal gases. These will serve as auxiliary calculation pathways for evaluating changes in state functions during irreversible processes. We use this procedure extensively in Chapter 13 on spontaneous processes and the second law of thermodynamics. 

Figure 1.5. Diagram of the change in state functions of a reaction that proceeds from state 1 (initial) to state 2 (final), showing the irrelevance of reaction pathway to AC, AH, and AS of the reaction.

The pressure (P) of an ideal gas or the volume (V) of water in a beaker are other examples of state functions. This path independence means that changes in state functions—AE, AP, and AV—depend only on their initial and final states. (Note that symbols for state functions, such as E, P, and V, are capitalized.)... 

Figure 1.5 Schematic of a quasi-static process compared with a finite irreversible process. The system initially in a state having properties T,-, P,-, and N, is to be changed to a final state having Ty, Pjr, and Ny. In the finite irreversible process (fop) the system passes through intermediate states that are undefined. During the quasi-static process (bottom) the change occurs in differential stages at the end of each stage the system is allowed to relax to an intermediate state that is well-defined. In both processes, overall changes in state functions, such as AT = Ty - T, and AP =...

Heat and work are not properties of either the system or the surroundings they exist only during the interaction that carries them across the boundary. However, for certain special processes Q and W are separately related to changes in state functions. We have already seen that if no thermal interaction exists, then the adiabatic work equals AH and it can be calculated assuming a reversible change. Likewise, if only a thermal interaction connects the system to the surroundings (the process is workfree), then the heat transferred equals AH,... 

Comment. Note that we have made a definitive statement about the feasibility of a proposed process without knowing details about the process itself. We are able to do so because the first and second laws effectively replace process-dependent heat and work effects with process-independent changes in state functions. 

Often we can simplify an analysis by combining the first and second laws to eliminate heat and work in favor of changes in state functions. Such replacements yield the Mw-damental equations of thermodynamics. These equations allow us to determine the effects of state changes without requiring us to evaluate heat and work. In what follows, we first present the forms for closed systems ( 3.2.1) and then give those for open systems ( 3.2.2). 

Exact differentials Changes in state functions Maxwell eqs. 

Changes In state functions do not depend on the pathway, but only on the initial and final state. 

Pressure (P), volume (V), and temperature (J) are some other state functions. Path independence means that changes in state functions—A , AP, AT, and AT—depend only on the initial and final states. 

The same relationship exists for most of the other state functions as well. (There is one exception, which we will see in the next chapter.) The differentials dw and dq are called inexact differentials, meaning that their integrated values w and q are path-dependent. By contrast, dU is an exact differential, meaning that its integrated value AC/is path-independent. All changes in state functions are exact differentials. 

Notice that is the heat that goes to the low-temperature reservoir in isothermal step 3 of the cycle, whereas qi is the heat that comes from the high-temperature reservoir in isothermal step 1. Each fraction therefore contains heat and temperatures from related parts of the universe under consideration. Because the other two steps are adiabatic (that is, qz = q4 = 0), equation 3.11 includes all of the heats of the Carnot cycle. The fact that these heats, divided by the absolute temperatures of the two reservoirs involved, add up to exactly zero is interesting. Recall that the cycle starts and stops at the same system conditions. But changes in state functions are dictated solely by the conditions of the system, not by the path that got the system to those conditions. If a system starts and stops at the same conditions, overall changes in state functions are exactly zero. Equation 3.11 suggests that/or reversible changes, a relationship between heat and absolute temperature is a state function. 

We have repeatedly made the point that some thermodynamic functions are state functions, and that changes in state functions are independent of the exact path taken. In other words, the change in a state function depends only on the initial and final conditions, not on how the initial conditions became the final conditions. 

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