Exact differential, defined

If the adiabatic work is independent of the path, it is the integral of an exact differential and suffices to define a change in a function of the state of the system, the energy U. (Some themiodynamicists call this the internal energy , so as to exclude any kinetic energy of the motion of the system as a whole.)... 

Pl.l Use the properties of the exact differential and the defining equations for the derived thermodynamic variables as needed to prove the following relationships ... 

In summary, the Carnot cycle can be used to define the thermodynamic temperature (see Section 2.2b), show that this thermodynamic temperature is an integrating denominator that converts the inexact differential bq into an exact differential of the entropy dS, and show that this thermodynamic temperature is the same as the absolute temperature obtained from the ideal gas. This hypothetical engine is indeed a useful one to consider. 

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. 

However, the cyclic integral of an exact differential is zero and therefore QJT is an exact differential of some function. The notation dQ, is used to emphasize that the process is reversible. The new function is called the entropy function and is defined in terms of its differential, so... 

Therefore, -S is a state property or an exact differential. Entropy cannot be easily defined but can be described in terms of entropy increase accompanying a particular process. 

This quantity is the exact differential of some state property and it is later defined as the entropy of the system ... 

The fact that the integral of dQ — dW around a closed path is zero is a necessary and sufficient condition that dQ — dWhc an exact differential form (Appendix). Thus, a function E can be defined such that... 

In this chapter we have developed the first and second laws for closed and open systems. For closed systems both laws are motivated by the desire to relate the process variables Q and W to changes in system properties. To emphasize this common theme, we have stated each law in two parts part 1 defines a new state function (either U or S) and part 2 imposes limitations on how the new state function changes with certain changes of state. For closed systems, the first law asserts that an exact differential (dU) is obtained from the algebraic sum of 8Q and 5W while the second law asserts that an exact differential (dS) is obtained by appl3fing an integrating factor to If a quantity forms an exact differential, then it is a system property, and changes in its value are not affected by the process that cormects two states. 

Since the quantities /of interest to us form exact differentials, the second-order variation in (G.0.5) is invariant under an exchange of the indices i and j. We need not proceed further into the variational calculus here because in this book we use the variations 8/and 8 /merely as a notational convenience relations (G.0.4) and (G.0.5) define the notation we use. 

Then (Sq/T) becomes a state function called entropy and T the absolute temperature. As a state function, entropy is path-independent. Eqn (1.25) is a mathematical statement of the second law of thermodynamics. The introduction of the integrating factor for 8q causes the thermal energy to be split into an extensive factor S and an intensive factor T. Clausius defined the entropy with the integrating factor of the inverse of absolute temperature in T 8q) = dS. Similarly, integrating factor 1/P in IP 6W) = dV leads to exact differential dV, which is formulated by Clapeyron in 1834. Introducing Eqn (1.25) into the first law of thermodynamics dU =8q + yields the combined first and second laws of thermodynamics... 

Any differential element of work, DW, is not an exact differential that is, Schwarz relations are not applicable, and integration from the initial to the final slate of the system depends on the path of integration (equation defining the particular process). 

The change in entropy, dS, is just QIT, where T is the temperature. The factor l/T is an integrating factor that transforms SQ into an exact differential just as 1/w transforms vdu — udv into the exact differential d(u/v). Because the change in entropy, dS = SQ/T,is exact differential, the change in entropy in a reversible cyclic process is zero. The entropy of a thermodynamic state is a well-defined single-valued function and the entropy is said to be a state function. An equivalent statement of the second law of thermodynamics is... 

This product PV has the status of a state function (it corresponds to a peculiar energy as will be discussed later) and therefore its variation must be given by an exact differential equation, thus requiring equality of the second derivatives (as explained earlier in Section 2.1.5 of Chapter 2). This defines their common value which is a second derivative of energy notated R" and has the... 

Heat capacity is thus defined as the rate of change of heat with temperature, dq not being an exact differential, the value of the heat capacity depends on the path of the process. 

Enthalpy H, is defined as the heat content of the system. It is an extensive property and dH is an exact differential ... 

From a mathematical standpoint the present proposition amounts to the demonstration that the thermodynamic temperature T is an integrating denominator for dg in a reversible change, i.e. dSs(dg/T) is an exact differential. The possibility of defining T depended, in its turn, on the use of Statement A, concerning the... 

Thus, even though heat is not a state function, and therefore dg is not an exact differential, the entropy S defined through Eq. (C.2) is a state function and d5 is an exact differential. Since the entropy is defined with reference to a reversible process, for an arbitrary process we will have... 

The differentials of these quantities are exact differentials. Work and heat are not state functions. There is no such thing as an amount of work or an amount of heat in a system. We have already seen that dwrev is not an exact differential. An infinitesimal amount of heat is also not an exact differential. For a system in which work is done only by changing the volume, the thermodynamic energy, U, is defined by its differential... 

Write the exact differential for any intensive thermodynamic property in terms of partial derivatives of specified independent, intensive properties. For example, given h= h T P), write dh. Define what is meant by independent properties and dependent properties. 

In thermodynamic applications the integral is often taken around a closed path. That is, the initial and final points in the x>y plane are identical. In this case the integral is equal to zero if the differential involved is exact, and different from zero if it is not. In mechanics the former condition defines what is called a conservative system (see Section 4.14). 

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