Thermodynamics constant-volume processes

In Chapter 3, we defined a new function, the internal energy U, and noted that it is a thermodynamic property that is, dU is an exact differential. As Q was defined in Equation (3.12) as equal to At/ when no work is done, the heat exchanged in a constant-volume process in which only PdV work is done is also independent of the path. For example, in a given chemical reaction carried out in a closed vessel of fixed volume, the heat absorbed (or evolved) depends only on the nature and condition of the initial reactants and of the final products it does not depend on the mechanism by which the reaction occurs. Therefore, if a catalyst speeds up the reaction by changing the mechanism, it does not affect the heat exchange accompanying the reaction. 

The first law of thermodynamics simply says that energy cannot be created or destroyed. With respect to a chemical system, the internal energy changes if energy flows into or out of the system as heat is applied and/or if work is done on or by the system. The work referred to in this case is the PV work defined earlier, and it simply means that the system expands or contracts. The first law of thermodynamics can be modified for processes that take place under constant pressure conditions. Because reactions are generally carried out in open systems in which the pressure is constant, these conditions are of greater interest than constant volume processes. Under constant pressure conditions Equation 3 can be rewritten as... 

Since the vessel size does not change appreciably, this situation maybe approximated by a constant volume process. Using the thermodynamic diagram, the initial volume is about 0.43 ft 3/lb. At this same volume and final temperature of 600 F, the pressure is ... 

One should not conclude from Eq 4.2-7 that the reversible work for any process is equal to the change in Helmholtz energy, since this result was derived only for an isothermal, constant-volume process. The value of VK , and the thermodynamic functions to which it is related, depends on the constraints placed on the system during the change of state (see Problem 4.3). For example, consider a process occurring in a closed system at fi.xed temperature and pressure. Here we have... 

The theoretical thermodynamic efficiency of an Otto-cycle engine is based on the efficiency of the constant volume process. Hence the engine s compression ratio and the specific heat ratio k of the working gas are relevant for this basic consideration ... 

Constant-pressure (or isobaric) processes occur under conditions where AP = 0. They are common in chemistry because reactions are often run in vessels that are open to the atmosphere, so the reaction proceeds at a constant pressure equal to that of the ambient atmospheric pressure. Constant-volume (or isochoric) processes, on the other hand, occur under conditions where AV = 0. If only expansion work is possible, the work done in a constant-volume process is zero (because w = —PextAV = 0). Combining this result with the first law of thermodynamics yields... 

Figure 7.8 (a) Thermodynamic cycle for pressure-volume changes, (b) Modelling one step of the cycle (continuous line) by using a fictitious constant pressure process followed by a fictitious constant volume process (dashed lines). 

Gases behave differently from liquids and must be characterized by an equation of state. Polytropic processes are studied in thermodynamics (Saad, 1966). Essentially, pv = constant, say C, where v is the specific volume also the index n of the proeess may vary from -oo to -too. For constant pressure processes, n = 0 for isothermal processes assuming perfect gases, n = 1. For reversible adiabatic processes, n = Cp/Cv, where Cp is the specific heat at constant pressure and Cv is the value obtained at constant volume. Finally, for constant volume processes, n = co. 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

This calculation shows that reaction energies and reaction enthalpies are usually about the same, even when reactions Involve gases. For this reason, chemists often use A 5 reaction nd A reaction interchangeably. Because many everyday processes occur at constant pressure, thermodynamic tables usually give values for enthalpy changes. Nevertheless, bear In mind that these are different thermodynamic quantities. For processes with modest AE values and significant volume changes, A " and A H can differ substantially. 

In thermodynamics, entropy enjoys the status as an infallible criterion of spontaneity. The concept of entropy could be used to determine whether or not a given process would take place spontaneously. It has been found that in a natural or spontaneous process there would be an increase in the entropy of the system. This is the most general criterion of spontaneity that thermodynamics offers however, to use this concept one must consider the entropy change in a process under the condition of constant volume and internal energy. Though infallible, entropy is thus not a very convenient criterion. There have, therefore, been attempts to find more suitable thermodynamic functions that would be of greater practical... 

Heat Content or Enthalpy. A thermodynamic property closely related to energy. It is defined by H = E + PV where E is the internal energy of the system, P is the pressure on the system and V is the volume of the system. Often it is used in differential form as in. AH = AE + PAV for a constant pressure process... 

The inequalities of the previous paragraph are extremely important, but they are of little direct use to experimenters because there is no convenient way to hold U and S constant except in isolated systems and adiabatic processes. In both of these inequalities, the independent variables (the properties that are held constant) are all extensive variables. There is just one way to define thermodynamic properties that provide criteria of spontaneous change and equilibrium when intensive variables are held constant, and that is by the use of Legendre transforms. That can be illustrated here with equation 2.2-1, but a more complete discussion of Legendre transforms is given in Section 2.5. Since laboratory experiments are usually carried out at constant pressure, rather than constant volume, a new thermodynamic potential, the enthalpy H, can be defined by... 

Heat capacities can be defined for processes that occur under conditions other than constant volume or constant temperature. For example, we could define a heat capacity at constant length of a sample. However, regardless of the nature of the process, the heat capacity will always be positive. This is ensured by the zeroth law of thermodynamics, which requires that as positive heat is transferred from a heat reservoir to a colder body, the temperature of the body will rise toward that of the reservoir in approaching the state of thermal equilibrium, regardless of the constraints of the heat-transfer process. 

Obviously, the energy functions of free energy F and free enthalpy G play the role of thermodynamic potentials for an irreversible process to occur in isothermal systems at constant volume and constant pressure, respectively. In general, the energy functions of F, and G can be used as the thermodynamic potentials to indicate the direction of an irreversible processes to occur under the condition that their respective characteristic variables remain constant. 

Thermochemistry is a branch of thermodynamics that deals with the change of heat (enthalpy) in chemical reactions. The heat absorbed or lost in chemical reactions usually occurs at constant pressure rather than at constant volume. The change of heat is mostly expressed by AH, the enthalpy change of a process from reactants to products ... 

Standard thermodynamic operations (Prigogine and Defay, 1954) on the Gibbs function, AG, yield expressions for related thermodynamic activation parameters. Thus the dependence of k on T can be used to calculate the enthalpy of activation, A, for processes at constant pressure or the thermodynamic energy of activation, A, for processes at constant volume, which in turn lead to the related entropies of activation, ASp and AS respectively. The dependence of k on pressure can be used to calculate the volume of activation, AV which is related to AHp by eqn (5) where a is the thermal... 

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