Abstract
Combining the First and the Second Law of Thermodynamics and choosing \(d\!\!^{-}\,W = P\,dV\) we find that
where μ is the so-called chemical potential. The inequality in (4.1) is valid in the course towards equilibrium and equality when equilibrium is reached. Thus, under the conditions \(T\,dS = P\,dV = dN = 0,\) the energy U is decreasing and reaches its minimum value when equilibrium is achieved. We can change independent variables from S to T and/or from V to P and/or from N to μ by subtracting respectively from both sides of (4.1) the differentials of the product TS, and/or adding the differentials of PV, and/or subtracting the differentials of μN. Thus various differentials of thermodynamic potentials result, e.g. that of the Gibbs free energy \(G \equiv U + PV - TS\) satisfying the relation \(dG \le - SdT + VdP + \mu dN\) which means that, under conditions of constant temperature and pressure and no exchange of matter, equilibrium corresponds to the minimum of G.
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- 1.
More accurately, U is the so-called internal energy, which is defined as the average value of the total energy \(E_{t}\) of the system, under conditions in which the total momentum is equal to zero on the average. (For macroscopic systems, the total angular momentum must also be zero on the average). Notice that the value of U or \(E_{t}\) is fully determined only after we choose a reference state, the energy of which is by definition zero. Usually, the reference state is the one in which all particles of the system are at infinite distance from each other and each one is in its ground state. There are three types of contributions to U or \(E_{t}\): The relativistic rest energy \(E_{o} = \sum {m_{oi} } c^{2}\), the kinetic energy \(E_{K} ,\) and the potential energy \(E_{P} ,\) which may include interactions both with the environment and among the particles of the system itself. For convenience, it is not uncommon to ignore the rest energy, if it remains constant, by incorporating it in the reference state.
- 2.
These inequalities are: \(T \ge 0,\,\,\,\,\,C_{V} > 0,\,\,\,\,\,C_{P} > \,C_{V} ,\,\,\,\,(\partial P/\partial V)_{T} < 0\) [3].
- 3.
A simple system, such as a perfect gas, when in equilibrium, can be described macroscopically by only three independent macroscopic variables, e.g. \(U,\,V,N;\) for a photon system in equilibrium N is not an independent variable and, hence, only two independent variables are sufficient for its macroscopic description (see Sect. 3.6). Other more complicated systems in equilibrium may require a larger number of independent macroscopic variables. Non-equilibrium macroscopic states of a system require more independent macroscopic variables than the ones required for the equilibrium state of the same system.
References
L.D. Landau, E.M. Lifshitz, Statistical Physics, Part 1 (Pergamon Press, Oxford 1980)
E. Fermi, Thermodynamics (Dover Publications, NY, 1956)
M.W. Zemansky, R.H. Dittman, Heat and Thermodynamics, 7th edn. (McGraw-Hill, New York, 1997)
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Economou, E.N. (2016). Equilibrium and Minimization of Total Energy. In: From Quarks to the Universe. Springer, Cham. https://doi.org/10.1007/978-3-319-20654-7_4
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DOI: https://doi.org/10.1007/978-3-319-20654-7_4
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