Internal Energy, U and Enthalpy

Application then of the First Law of Thermodynamics (Frames 1, 2 and 8) leads to the relationship  

Where a thermodynamic system has to a fixed volume, A V = 0 (e.g. in a bomb calorimeter or autoclave) and not permitted to expand (w = 0) and then  

We note here, as was found in the example of the calculation of the work done during the irreversible adiabatic of expansion of a gas (equation 9.4 Frame 9), that under specific conditions a path dependent function can become identically equal to the change in a state function. 

In order to describe the heat changes, qP which take place in a system (where the external pressure P is constant - most usually being equal to Patm (= 1 bar)) we need to rearrange equation (10.1), since in this system work of expansion (leading to a change of volume) may now be permitted, so that  

Substitution of equation (10.4) and (10.5) into (10.3) and grouping terms, leads us to  

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Expressions for internal energy, U, and enthalpy, H, in terms of pressure (P), volume (L), and absolute temperature (P) ... 

Some aspects of the kinetic molecular theory (KMT) of ideal gases were outlined in Sidebar 2.7. The simplest form of KMT refers to monatomic ideal gases, for which the internal energy U and enthalpy H=U + PV = U + nRT can be written explicitly as... 

The differential changes in the internal energy u and enthalpy h of an ideal gas can be expressed in terms of the specific heats as... 

In this chapter, a mathematical expression of the First Law will be discussed. The two functions internal energy, U, and enthalpy, H, will figure prominently. In addition, tabulations of standard enthalpy changes of formation, A//0 for a number of compounds, will be given. From such tables it is possible to derive the amount of thermal energy associated with any reaction, as long as all reactants and products are listed. 

Consider an ideal gas composed of diatomic molecules AB. In the limit of absolute zero temperature, all the AB molecules are in their ground states of electronic and nuclear motion, so DqN/ (where is the Avogadro constant and Dq is for the ground electronic state of AB) is the change in the thermodynamic interned energy U and enthalpy H for dissociation of 1 mole of ideal-gas diatomic molecules N/>,Dq = AUl = A//°o for AB(g) A(g) + B(g). 

In fact, heat capacities are commonly not defined in terms of exchanged heat (Qe), but are directly used as derivatives of internal energy U and enthalpy //. The disadvantage here is that, for every side condition (constant volume, constant pressure, constant X, etc.), a different quantity is necessary for the role of heat content. ... 

Classical thermodynamics describes states of equilibrimn and processes that connect states of equilibrium. State functions such as internal energy U and enthalpy H, which are independent of the process path in a change of state, are used. 

In most applications, thermodynamics is concerned with five fundamental properties of matter volume (V), pressure (/ ), temperature (T), internal energy (U) and entropy (5). In addition, three derived properties that are combinations of the fundamental properties are commonly encountered. The derived properties are enthalpy (//). Helmholtz free energy (A) and Gibbs free energy ) ... 

Here, h = u + P/p is the enthalpy per unit mass of fluid. Note that the inlet and exit streams include enthalpy (i.e., both internal energy, u, and flow work, P/p), whereas the system energy includes only the internal energy but no P/p flow work (for obvious reasons). If there are only one inlet stream and one exit stream (m, =m0 = m) and the system is at steady state, the energy balance becomes... 

But not all of the heater s energy q goes into raising U. We need some of it to perform pressure-volume work, since the vapour formed on boiling works to push back the external atmosphere. The difference between the internal energy U and the available energy (the enthalpy) is given by... 

Derivation of the expression for the minimum production of S in the systems with constant T and V (volume) differs from the one above only by replacement of enthalpy by internal energy (U) and the Gibbs energy by the Helmholtz energy in the equations. When we set S and P or S and V dissipation turns out to be zero according to the problem statement. In the case of constant U and V or H and P, the interaction with the environment does not hinder the relaxation of the open subsystem toward the state max Sos. 

Enthalpy (H) An extensive property of a substance that can be used to obtain the heat absorbed or released by a chemical reaction or physical change at constant pressure. It is defined as the sum of the internal energy (U) and the product of the pressure and the volume of the system (PV) H = U + PV. 

The internal energy U and the enthalpy // of a closed homogeneous system of constant composition are described by the internal state variables p, V and T which are related to each other via the equation of state. 

The internal energy U, the enthalpy H, the free internal energy F, and the free enthalpy G, must be at a minimum for a system to be at equilibrium. This is comparable to a me-... 

These equations show that for an ideal gas the internal energy u, the enthalpy h, and the specific heat capacities at constant pressure Cp and at constant volume are only functions of the temperature and independent of pressure or volume. On the other hand, the specific entropy (s), the specific Gibbs energy (g), and the specific Helmholtz energy (o) are functions of temperature and pressure or volume even for an ideal gas. 

The specific internal energy u and the specific enthalpy h are related by h = u + Pv... 

We cannot measure the absolute internal energy U or enthalpy H because the zero of energy is arbitrary. As a result, we are usually only interested in determining changes in these properties (At/ and A.H) during a process. However, it is possible to determine the absolute entropy of a substance. This is because of the third law of thermodynamics, which states that the entropy of a pure substance in its thermodynamically most stable form is zero at the absolute zero of temperature, independent of pressure. For the vast majority of substances, the thermodynamically most stable form at 0 K is a perfect crystal. An important exception is helium, which remains liquid, due to its large quantum zero-point motion, at 0 K for pressures below about 10 bar. 

Offhand, it may seem surprising that the internal energy U and the enthalpy H of ideal gas only depend on the gas temperature T, i.e. that irrespective of the gas pressure p and volume V, all ideal gases have the same U and H, respectively, at the same temperature. 

In applying the first law, two important state functions have been introduced and defined, namely the internal energy U and the enthalpy H. The second law introduces and defines a new fundamental state function entropy S. Before introducing this new state function, however, it is necessary to define some previously mentioned concepts. 

Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U enthalpy, H and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows ... 

By mathematical manipulation, numerous additional relationships can be derived from those given in Table 2-19. Of particular significance are expressions that relate enthalpy H and internal energy U to the measurable variables, P, V, and T. Thus, choosing the basis as one pound mass,... 

The essential changes with respect to the canonical ensemble is the substitution of the internal energy U by the enthalpy H = U + PV in equations (7) and (8). 

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