Enthalpy internal energy relationship

Once we have the enthalpy, internal energy is calculated from its relationship to H ... 

To use Equation 6-12, a relationship is required between either the internal energy per unit mass u or the enthalpy per unit mass h and the state variables such as the temperature T, pressure P, and composition nij. 

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,... 

Statistical thermodynamics provides the relationships that we need in order to bridge this gap between the macro and the micro. Our most important application will involve the calculation of the thermodynamic properties of the ideal gas, but we will also apply the techniques to solids. The procedure will involve calculating U — Uo, the internal energy above zero Kelvin, from the energy of the individual molecules. Enthalpy differences and heat capacities are then easily calculated from the internal energy. Boltzmann s equation... 

It is thus seen that heat capacity at constant volume is the rate of change of internal energy with temperature, while heat capacity at constant pressure is the rate of change of enthalpy with temperature. Like internal energy, enthalpy and heat capacity are also extensive properties. The heat capacity values of substances are usually expressed per unit mass or mole. For instance, the specific heat which is the heat capacity per gram of the substance or the molar heat, which is the heat capacity per mole of the substance, are generally considered. The heat capacity of a substance increases with increase in temperature. This variation is usually represented by an empirical relationship such as... 

This relationship, of course, will hold for a shock wave when q is set equal to zero. The Hugoniot equation is also written in terms of the enthalpy and internal energy changes. The expression with internal energies is particularly useful in the actual solution for the detonation velocity tq. If a total enthalpy (sensible plus chemical) in unit mass terms is defined such that... 

A very important problem in the thermodynamics of deformation of condensed systems is the relationship between heat and work. From Eqs. (2) and (4) by integration, the internal energy and enthalpy can be derived. As in other condensed systems, the enthalpy differs from the internal energy at atmospheric pressure only negligibly, since the internal pressure in condensed systems P > P. Therefore, the work against the atmospheric pressure can be neglected in comparison with the term jX.. Hence it follows that... 

To perform energy balance calculations on a reactive system, proceed much as you did for nonreactive systems (a) draw and label a flowchart (b) use material balances and phase equilibrium relationships such as Raoult s law to determine as many stream component amounts or flow rates as possible (c) choose reference states for specific enthalpy (or internal energy) calculations and prepare and fill in an inlet-outlet enthalpy (or internal energy) table and (d) calculate AH (or AC/ or A/C), substitute the calculated value in the appropriate form of the energy balance equation, and complete the required calculation. 

AH is positive when heat is supplied to a system which is free to change its volume and negative when the system releases heat (as in an exothermic reaction). Enthalpy is related to the internal energy of a system by the relationship... 

The internal energy per mole of a chemical system is the sum of all energies, electronic, vibrational, and kinetic, possessed by one mole of molecules at a given temperature, pressure and volume. Its thermodynamic symbol is U, and the relationship to enthalpy, H, is given by H = U + PV. Chemists use enthalpies because for any process, dFi is equal to the heat exchanged, dq, without contributions from volume work. 

We may note that the energy conservation principle (or, equivalently, the first law of thermodynamics) has not improved the balance between the number of unknown, independent variables and differential relationships between them. Indeed, we have obtained a single independent scalar equation, either (2 47 ) or (2-51), but have introduced several new unknowns in the process, the three components of q and either the specific internal energy e or enthalpy h. A relationship between e or h and the thermodynamic state variables, say, pressure p and temperature 9, can be obtained provided that equilibrium thermodynamics is assumed to be applicable to a fluid element that moves with a velocity u. In particular, a differential change in 9 orp leads to a differential change in h for an equilibrium system ... 

This simply shows that there is a physical relationship between different quantities that one can measure in a gas system, so that gas pressure can be expressed as a function of gas volume, temperature and number of moles, n. In general, some relationships come from the specific properties of a material and some follow from physical laws that are independent of the material (such as the laws of thermodynamics). There are two different kinds of thermodynamic variables intensive variables (those that do not depend on the size and amount of the system, like temperature, pressure, density, electrostatic potential, electric field, magnetic field and molar properties) and extensive variables (those that scale linearly with the size and amount of the system, like mass, volume, number of molecules, internal energy, enthalpy and entropy). Extensive variables are additive whereas intensive variables are not. 

The relationship between the change in internal energy and change in enthalpy of the system for a reaction at a given temperature is given by the equation ... 

A question that immediately comes to mind is what might be the relationship between this enthalpy of activation and the quantity we have been calling activation energy. If we let the latter play the role of internal energy in classical thermodynamics, then... 

Another useful result is obtained by expressing the relationship between enthalpy and internal energy in the ideal-gas state. Starting with eg. fr. i2l and using Vs = RT/P, we have... 

It is not necessary to derive equations for any other residual properties because these can all be related to Vr, Hr, and Sr. In general, all relationships among regular properties can also be written among the corresponding residual properties. For example, internal energy is related to enthalpy through the relationship,... 

Taking the difference between the two, we obtain the relationship between the residual enthalpy and residual internal energy ... 

The energy contents of an enclosed pressure liquefied gas is determined by its internal energy, u(T,v). The corresponding values can be read from tables. These mostiy only fist enthalpies, h(T,v), so that the following relationship has to be used... 

Free (Gibbs free) Enthalpy (thermodynamic potential), G The relationship between the enthalpy H, entropy S and internal energy U is defined as... 

First Law of Thermodynamics The first law of thermodynamics, which is based on the law of conservation of energy, relates the internal energy change of a system to the heat change and the work done. It can also be expressed to show the relationship between the internal energy change and enthalpy change of a process. 

In the preceding discussion, we noted that the enthalpy change equals the heat of reaction at constant pressure. This will be sufficient for the purpose of this chapter, which is to introduce the concepts of heat of reaction and enthalpy change. Later we will look at thermodynamics in more detail. Still, it is useful at this point to note briefly the relationship of enthalpy to internal energy. 

From the first-law relationship between the internal energy and H, the enthalpy ... 

The result of these simplifications is a balance for the internal energy. From thermodynamics it is known that the enthalpy H is the sum of the internal energy and the product PV. Using this thermodynamic relationship ... 

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