next up previous
Next: Thermodynamic Stability Up: Maximal Work, Minimal Work Previous: Maximal Work, Minimal Work

Subsections


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).
\begin{figure*}
 \InputIfFileExists{unequilibr.pstex_t}{}{}\end{figure*}
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,  
 \begin{displaymath}
 R = E_{\text{start}}-E_{\text{fin}}\end{displaymath} (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):

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

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:
\begin{figure*}
 \InputIfFileExists{environ.pstex_t}{}{}\end{figure*}
What is the minimal work to change the state of the body $R_{\min}$?

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

We obtained:

\begin{displaymath}
\Delta E = R + P_0\,\Delta V_0 - T_0\,\Delta S_0\end{displaymath}

Result:  
 \begin{displaymath}
 \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}\end{displaymath} (2)
Particular cases: 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:

\begin{displaymath}
\Delta( E - T_0 S + P_0 V)\le 0\end{displaymath}

This thing always decreases!

Probability of a Fluctuation

Once again, a body in an environment (microcanonical ensemble):
\begin{figure*}
 \InputIfFileExists{environ.pstex_t}{}{}\end{figure*}
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

\begin{displaymath}
\Prob\sim\exp(\Delta S/k)\end{displaymath}

In equilibrium

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

When the system equilibrates, its total entropy increases, so

\begin{displaymath}
S_{\text{start}}<S_{\text{eq}}(E_{\text{tot}})\end{displaymath}

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

\begin{displaymath}
S_{\text{eq}}(E_{\text{eq}}) = S_{\text{start}}\end{displaymath}


\begin{figure*}
 \InputIfFileExists{ST.pstex_t}{}{}\end{figure*}
Then:

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

Therefore

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

We obtained:  
 \begin{displaymath}
 \Prob\sim\exp(-R_{\min}/kT)\end{displaymath} (3)
Difference between (3) and Gibbs formulae: here we discuss the probability of a macroscopic state!


next up previous
Next: Thermodynamic Stability Up: Maximal Work, Minimal Work Previous: Maximal Work, Minimal Work

© 1997 Boris Veytsman and Michael Kotelyanskii
Fri Sep 12 00:09:21 EDT 1997