Temperature exact differential

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

Equation (2.18) is another example of a line integral, demonstrating that 6q is not an exact differential. To calculate q, one must know the heat capacity as a function of temperature. If one graphs C against T as shown in Figure 2.8, the area under the curve is q. The dependence of C upon T is determined by the path followed. The calculation of q thus requires that we specify the path. Heat is often calculated for an isobaric or an isochoric process in which the heat capacity is represented as Cp or Cy, respectively. If molar quantities are involved, the heat capacities are C/)m or CY.m. Isobaric heat capacities are more... 

We have previously shown that the Pfaff differential <5pressure-volume work equation (2.43) is an inexact differential. It is easy to show that division of equation (2.43) by the absolute temperature T yields an exact differential expression. The division gives... 

To summarize, the Carnot cycle or the Caratheodory principle leads to an integrating denominator that converts the inexact differential 8qrev into an exact differential. This integrating denominator can assume an infinite number of forms, one of which is the thermodynamic (Kelvin) temperature T that is equal to the ideal gas (absolute) temperature. The result is... 

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. 

An infinitesimal change in internal energy is an exact differential and is a unique function of temperature and pressure (for a given composition). Since the density of a given material is also uniquely determined by temperature and pressure (e.g., by an equation of state for the material), the internal energy may be expressed as a function of any two of the three terms T, P, or p (or v = 1/p). Hence, we may write ... 

The thermodynamic changes for reversible, free, and intermediate expansions are compared in the first column of Table 5.1. This table emphasizes the difference between an exact differential and an inexact differential. Thus, U and H, which are state functions whose differentials are exact, undergo the same change in each of the three different paths used for the transformation. They are thermodynamic properties. However, the work and heat quantities depend on the particular path chosen, even though the initial and final values of the temperature, pressure, and volume, respectively, are the same in all these cases. Thus, heat and work are not thermodynamic properties rather, they are energies in transfer between system and surroundings. 

Example 5.2 Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature Exact Solution The semi-infinite solid in Fig. E5.2 is initially at constant temperature Tq. At time t — 0 the surface temperature is raised to T. This is a one-dimensional transient heat-conduction problem. The governing parabolic differential equation... 

To calculate the enthalpy per unit mass, we use the property that the enthalpy is also an exact differential. For a pure substance, the enthalpy for a single phase can be expressed in terms of the temperature and pressure (a more convenient variable than specific volume) alone. If we let... 

Testing of the temperature dependence of the extension and shrinkage behaviour at low (TMA) or oscillating (DMA) yarn tension glass temperature, elasticity and other parameters for exact differentiation of elastomeric fibres. 

The statement that the functions U and S are independent of path has the physical significance that the functions U and S depend on the state of a system as described by its temperature, pressure, and composition, and they are independent of all previous states of the system. That is, such functions are independent of the history of the system, and they are known by a variety of names such as potential functions, exact differentials, and point functions. ... 

In thermodynamics, quantities such as the thermodynamic energy, the volume, the pressure, the temperature, the amount of substances, and so forth, are functions of the variables that can be used to specify the state of the system. They are called state functions. 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 in Example 7.11 that dwrev is not an exact differential, and the same is true of dw for an irreversible process. An infinitesimal amount of heat is also not an exact differential. However, the first law of thermodynamics states that dU, which equals dq -H dw, is an exact differential. 

The entropy of ideal dissolution (10.9) can also be easily derived using classical (as opposed to statistical) thermodynamics. This is worth doing here since it provides further insight into the problem. The derivation for ideal gases is very simple, and that for liquids and solids only slightly more complicated. Because we want to look at the effect of volume and pressure changes at constant temperature, we start with the exact differential of S with respect to T and V,... 

The differential dU is exact, depending only on the initial and final states (pressure, volume, temperature, composition). Contrary, both SQ and SW are not exact differentials, depending on the path of transformation. 

Since is an exact differential, second mixed partial derivatives of are independent of the order in which differentiation is performed. This leads to a Maxwell relation between the temperature dependence of (Pa and the mass fraction dependence of specific entropy s. Hence,... 

The existence of states that are inaccessible to adiabatic processes was shown by Carath odory to be necessary and sufficient for the existence of an integrating factor that converts into an exact differential [2-4]. From the calculus we know that for differential equations in two independent variables, an integrating factor always exists in fact, an infinite number of integrating factors exist. Experimentally, we find that for pure one-phase substances, only two independent intensive properties are needed to identify a thermod)mamic state. So for the experimental situation we have described, we can write SQ gj, as a function of two variables and choose the integrating factor. The simplest choice is to identify the integrating factor as the positive absolute thermodynamic temperature X = T. Then (2.3.3) becomes... 

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... 

Equation (49a) is the exact differential of quantity Z [Eq. (3)] at constant pressure and temperature. Equation (49b) is a Gibbs-Duhem-Margules type of equation, indicating the mutual dependence of partial molar quantities. 

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... 

A somewhat more mathematical way to express this analogy between work-pressure-volume and heat-temperature-entropy, and which is discussed in more detail in C.2.1, is to note that although (or if you prefer that notation) is not the differential of any function (not an exact differential) i rev/-P is, being equal to -Ay, or -dV in differential notation. So one might suppose that similarly although is not exact, perhaps is. And indeed it is, being equal to dS. 

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