Subsections

• Maximal Work
• System and Environment
• Thermodynamic Potentials and Irreversible Processes
• Probability of a Fluctuation

Maximal and Minimal Work

Maximal Work

So far we discussed reversible processes. What happens for irreversible ones?

Let us discuss an insulated system (adiabatic process).

We know the volume of the system in the final state $V_{\text{fin}}$. What is the maximal work R against the surroundings?

Since there is no heat exchange,  

$$R = E_{\text{start}}-E_{\text{fin}}$$ (1)

$E_{\text{start}}$ is given. We want $E_{\text{fin}}$ to be as small as possible (we are greedy!). Since in final state the system is in equilibrium, $E_{\text{fin}}=E_{\text{fin}}(S_{\text{fin}})$. Differentiate (1):

$$\left(\frac{\partial R}{\partial S_{\text{fin}}}\right)_V = ... ...al E_{\text{fin}}}{\partial S_{\text{fin}}}\right)_V = -T < 0$$

R is a decreasing function of $S_{\text{fin}}$. The lower is $S_{\text{fin}}$, the better! But S cannot decrease, it can only increase!

Conclusion:

R in a closed system is maximal for reversible processes, when $S=\mathit{const}$

Machines should produce as less entropy as possible!

System and Environment

Consider a not insulated body at temperature T and pressure P surrounded by environment with temperature T₀ and pressure P₀:

What is the minimal work to change the state of the body $R_{\min}$?

• If the source makes work R, the body makes work -R
• Minimal work of the source corresponds to maximal work of the body

What is the change of energy $\Delta E$ of the body?

• Work from the external source R
• Work from the environment $R_{in}=P_0\,\Delta V_0$
• Heat from the environment $-T_0\,\Delta S_0$

We obtained:

$$\Delta E = R + P_0\,\Delta V_0 - T_0\,\Delta S_0$$

• Since $V+V_0=\mathit{const}$, $\Delta V= - \Delta V_0$
• For reversible processes $\Delta S_{\text{tot}}= \Delta S + \Delta S_0 =0$, for irreversible processes $\Delta S_{\text{tot}} = \Delta S + \Delta S_0 \gt 0$. So $\Delta S_0 \ge - \Delta S$

Result:  

$$\begin{aligned} R \ge R_{\min} &= \Delta E - T_0\,\Delta S + P_0\, \Delta V =\ &\Delta( E - T_0 S + P_0 V) \end{aligned}$$ (2)

Particular cases:

• T=T₀, $\Delta V=0$ (a non-insulated beaker):

  $$\Delta R_{\min} = \Delta(E-TS) = \Delta A$$

• T=T₀, P=P₀ (a non-insulated beaker under flexible stopper):

  $$\Delta R_{\min} = \Delta(E-TS + PV) = \Delta G$$

• P=P₀, $\Delta S=0$ (an insulated beaker under flexible stopper)

  $$\Delta R_{\min} = \Delta(E+PV) = \Delta H$$

Thermodynamic potentials describe the minimal work to produce the given state at certain external conditions.

Thermodynamic Potentials and Irreversible Processes

Suppose now R=0. We obtained:

$$\Delta( E - T_0 S + P_0 V)\le 0$$

This thing always decreases!

• If T=T₀, $V=\mathit{const}$, Helmholtz free energy A decreases,
• If T=T₀, P=P₀, Gibbs free energy G decreases,
• If P=P₀, $S=\mathit{const}$, enthalpy H decreases.

Probability of a Fluctuation

Once again, a body in an environment (microcanonical ensemble):

Let us consider it as a fluctuation: some macroscopic variable deviated from the mean value. Then $R_{\min}$ is the minimal work to create the fluctuation.

Probability for the fluctuation is

$$\Prob\sim\exp(\Delta S/k)$$

In equilibrium

$$S_{\text{eq}}=S_{\text{eq}}(E_{\text{tot}}), \quad E_{\text{tot}} = E+ E_0$$

When the system equilibrates, its total entropy increases, so

$$S_{\text{start}}<S_{\text{eq}}(E_{\text{tot}})$$

Let $E_{\text{eq}}$ be the energy for which

$$S_{\text{eq}}(E_{\text{eq}}) = S_{\text{start}}$$

Then:

$$R_{\min}= E_{\text{tot}}-E_{\text{eq}},\quad \Delta S = S_{\text{eq}}(E_{\text{tot}}) - S_{\text{start}}$$

Therefore

$$\Delta S = - \frac{dS_{\text{eq}}}{dE_{\text{tot}}}R_{\min} = -\frac{R_{\min}}{T_0}$$

We obtained:  

$$\Prob\sim\exp(-R_{\min}/kT)$$ (3)

Difference between (3) and Gibbs formulae: here we discuss the probability of a macroscopic state!

• Fluctuations at $T=\mathit{const}$, $V=\mathit{const}$: $\Prob\sim\exp(-\Delta A/kT)$
• Fluctuations at $T=\mathit{const}$, $P=\mathit{const}$: $\Prob\sim\exp(-\Delta G/kT)$
• Fluctuations at $P=\mathit{const}$, $S=\mathit{const}$ $\Prob\sim\exp(-\Delta H/kT)$