## Helmholtz free energy – wikipedia gas 2015

###########

In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature and volume ( isothermal, isochoric). The gas tax negative of the change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which volume is held constant. If the volume were not held constant, part of this work would be performed as boundary work. This makes the Helmholtz energy useful for systems held at constant volume. Furthermore, at constant temperature, the Helmholtz energy is minimized at equilibrium.

In contrast, the Gibbs free energy or free enthalpy is most commonly used as a measure of thermodynamic potential (especially in chemistry) when it is convenient for applications that occur at constant pressure. For example, in explosives research Helmholtz free energy is often used, since explosive reactions by their power generation definition nature induce pressure changes. It is also frequently used to define fundamental equations of state of pure substances.

The concept of free energy was developed by Hermann von Helmholtz, a German physician and physicist, and first presented in 1882 in a lecture called On the thermodynamics of chemical processes.  From the German word Arbeit (work), the International Union of Pure and Applied Chemistry (IUPAC) recommends the symbol A and the name Helmholtz energy.  In physics, the symbol F is also used in reference to free energy or Helmholtz function.

where U {\displaystyle U} is the internal energy, δ Q {\displaystyle \delta Q} is the energy added as heat, and δ W {\displaystyle \delta W} is the work done on the system. The second law of thermodynamics for a reversible process yields δ Q = T d S {\displaystyle \delta Q=T\,\mathrm {d} S} . In case of a reversible change, the work done can be expressed as δ W = − p d V {\displaystyle \delta W=-p\,\mathrm {d} V} (ignoring electrical and other non- PV work):

The laws of thermodynamics are most easily applicable to systems undergoing reversible processes or processes that begin and end in thermal equilibrium, although irreversible quasistatic processes gas mask art or spontaneous processes in systems with uniform temperature and pressure (u PT processes) can also be analyzed  based on the fundamental thermodynamic relation as shown further below. First, if we wish to describe phenomena like chemical reactions, it may be convenient to consider suitably chosen initial and final states in which the system is in (metastable) thermal equilibrium. If the system is kept at fixed volume and is in contact with a heat bath at some constant temperature, then we can reason as follows.

Since the thermodynamical variables of the system are well defined in the initial state and the final state, the internal energy increase Δ U {\displaystyle \Delta U} , the entropy increase Δ S {\displaystyle \Delta S} , and the total amount of work that can be extracted, performed by the system, W {\displaystyle W} , are well defined quantities. Conservation of energy implies

This result seems to contradict the equation d F = − S d T − P d V, as keeping T and V constant seems to imply d F = 0, and hence F = constant. In reality there is no contradiction: In a simple one-component system, to which the validity of the equation d F = − S d T − P d V is restricted, no process can occur at constant T and V, since there is a unique P( T, V) relation, and thus T, V, and P are all fixed. To allow grade 9 electricity test and answers for spontaneous processes at constant T and V, one needs to enlarge the thermodynamical state space of the system. In case of a chemical reaction, one must allow for changes in the numbers electricity video ks1 N j of particles of each type j. The differential of the free energy then generalizes to

where the N j {\displaystyle N_{j}} are the numbers of particles of type j, and the μ j {\displaystyle \mu _{j}} are the corresponding chemical potentials. This equation is then again valid for both reversible and non-reversible u PT  changes. In case of a spontaneous change at constant T and V without electrical work, the last term will thus be negative.

where c is some constant. The value of c can be determined by considering the limit T → 0. In this limit the entropy becomes S = k log ⁡ Ω 0 {\displaystyle S=k\log \Omega _{0}} , where Ω 0 {\displaystyle \Omega _{0}} is the ground-state degeneracy. The partition function in this limit is Ω 0 e − β U 0 {\displaystyle \Omega _{0}e^{-\beta U_{0}}} , where U 0 {\displaystyle U_{0}} is the ground-state energy. Thus, we see that c = 0 {\displaystyle c=0} and that

one can find expressions for entropy, pressure and chemical potential:  S = − ( ∂ F ∂ T ) | V , N , P = − ( ∂ F ∂ V ) | T , N , μ = ( ∂ F ∂ N ) | T , V . {\displaystyle S=-{\bigg (}{\frac {\partial gas station car wash F}{\partial T}}{\bigg )}{\bigg |}_{V,N},\quad P=-{\bigg (}{\frac {\partial F}{\partial V}}{\bigg )}{\bigg |}_{T,N},\quad \mu ={\bigg (}{\frac {\partial F}{\partial N}}{\bigg )}{\bigg |}_{T,V}.}

We can easily generalize this proof to the case of quantum mechanical models. We denote the eigenstates of H ~ {\displaystyle {\tilde {H}}} by | r ⟩ {\displaystyle \left|r\right\rangle } . We denote the diagonal components of the density matrices for the canonical distributions for H {\displaystyle H} and H ~ {\displaystyle {\tilde {H}}} in this basis as:

In the more general case, the mechanical term p d V {\displaystyle p\mathrm {d} V} must be replaced by the product of volume, stress, and an infinitesimal strain:  d F = V ∑ i j σ i j d ε i j − S d T + ∑ i μ i d N i , {\displaystyle \mathrm {d} F=V\sum _{ij}\sigma _{ij}\,\mathrm {d} \varepsilon _{ij}-S\,\mathrm {d} T+\sum _{i}\mu _{i}\,\mathrm {d} N_{i},}

F = 1 2 V C i j k l ε i j ε k l − S T + ∑ i μ i N i = 1 2 V σ i j ε i j − S T + ∑ i μ i N i . {\displaystyle {\begin{aligned}F={\frac {1}{2}}VC_{ijkl}\varepsilon _{ij}\varepsilon _{kl}-ST+\sum _{i}\mu _{i}N_{i}\\={\frac {1}{2}}V\sigma _{ij}\varepsilon _{ij}-ST+\sum _{i}\mu _{i}N_{i}.\end{aligned}}} Application to fundamental equations of state [ edit ]

Hinton and Zelem  derive an objective function for training auto-encoder based on the minimum description length (MDL) principle. The description length of an input vector using a particular code is the sum of the code cost and reconstruction cost. [They] define this to be the energy of the code, for reasons that will become clear later. Given an input vector, [they] define the energy electricity and circuits class 6 pdf of a code to be the sum of the code cost and the reconstruction cost. The true expected combined cost is