Simple system internal energy

We can see how the values of heat capacities depend on molecular properties by using the relations in Section 6.7. We start with a simple system, a monatomic ideal gas such as argon. We saw in Section 6.7 that the molar internal energy of a monatomic ideal gas at a temperature T is RT and that the change in molar internal energy when the temperature is changed by AT is A(Jm = jRAT. It follows from Eq. 12a that the molar heat capacity at constant volume is... 

It is noted that all systems in turmoil tend to subside spontaneously to simple states, independent of previous history. It happens when the effects of previously applied external influences damp out and the systems evolve toward states in which their properties are determined by intrinsic factors only. They are called equilibrium states. Experience shows that all equilibrium states are macroscopically completely defined by the internal energy U, the volume V, and the mole numbers Nj of all chemical components. 

Complete potential energy surfaces have been developed for a few very simple systems. However, they will remain scarce in the future for two primary reasons. First, die geometry of an N-atom molecule is described by 3N-6 internal coordinates. If the energy of 10 different values along each of these internal coordinates is sufficient to describe the surface, then 103N 6 energy points still must be computed. [101] For benzene, where N = 12, this is 1030 energy points Even with the capability... 

The zeroth law of thermodynamics involves some simple definition of thermodynamic equilibrium. Thermodynamic equilibrium leads to the large-scale definition of temperature, as opposed to the small-scale definition related to the kinetic energy of the molecules. The first law of thermodynamics relates the various forms of kinetic and potential energy in a system to the work which a system can perform and to the transfer of heat. This law is sometimes taken as the definition of internal energy, and introduces an additional state variable, enthalpy. 

A thermodynamic system is a part of the physical universe with a specified boundary for observation. A system contains a substance with a large amount of molecules or atoms, and is formed by a geometrical volume of macroscopic dimensions subjected to controlled experimental conditions. An ideal thermodynamic system is a model system with simplifications to represent a real system that can be described by the theoretical thermodynamics approach. A simple system is a single state system with no internal boundaries, and is not subject to external force fields or inertial forces. A composite system, however, has at least two simple systems separated by a barrier restrictive to one form of energy or matter. The boundary of the volume separates the system from its surroundings. A system may be taken through a complete cycle of states, in which its final state is the same as its original state. 

An extremum principle minimizes or maximizes a fundamental equation subject to certain constraints. For example, the principle of maximum entropy (dS)v = 0 and, (d2S)rj < 0, and the principle of minimum internal energy (dU)s = 0 and (d2U)s>0, are the fundamental principles of equilibrium, and can be associated with thermodynamic stability. The conditions of equilibrium can be established in terms of extensive parameters U and. S, or in terms of intensive parameters. Consider a composite system with two simple subsystems of A and B having a single species. Then the condition of equilibrium is... 

For the entropy and internal energy, the canonical variables consist of extensive parameters. For a simple system, the extensive properties are S, U, and V. and the fundamental equations define a fundamental surface of entropy S = S(U,V) in the Gibbs space of S, U, and V. 

Both operators, i.e., the total Hamiltonian, Htot, and the kinetic energy of the CM motion, TCm, have simple forms in the Cartesian coordinate system. The full optimization effort can now be directed solely to improving the internal energy of the system because the functional (15) now contains only the internal Hamiltonian. One can expect that after optimization the variational wave function will be a sum of products of the integral ground state and wave functions representing different states of the CM motion ... 

To understand robber elasticity we have to revisit some simple thermodynamics (the horror. the horror ). Let s start with the Helmholtz free energy of our piece of rubber, by which we mean that we are considering the free energy at constant temperature and volume (go to the review at the start of Chapter 10 if you ve also forgotten this stuff). If E is the internal energy (the sum of the potential and kinetic energies of all the particles in the system) and 5 the entropy, then (Equation 13-26) ... 

In chemical process units such as reactors, distillation columns, evaporators, and heat exchangers, shaft work and kinetic and potential energy changes tend to be negligible compared with heat flows and internal energy and enthalpy changes. Energy balances on such units therefore usually omit the former terms and so take the simple form Q = U (closed system) or Q = AH (open system),... 

Equation 11.3-7 is simple in appearance, but its solution is still generally difficult to obtain. If, for example, the composition or temperature of the system contents varies with position in the system, it is difficult to express the total internal energy t/sys in terms of measurable quantities, and a similar problem occurs if phase changes or chemical reactions take place in the course of the process. To illustrate the solution of energy balance problems without becoming too involved in the thermodynamic complexities, we will impose the additional restrictions that follow. 

This definition accommodates both the molar heat capacity and the specific heat capacity (usually called specific heat), depending on whether U is the molar or specific internal energy. Although this definition makes no reference to any process, it relates in an especially simple way to a constant-volume process in a closed system, for which Eq. (2.16) may be written ... 

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