148:(and fixed external parameters), the second law states that the entropy will increase to a maximum value at equilibrium. An isolated system has a fixed total energy and mass. A closed system, on the other hand, is a system which is connected to another, and cannot exchange matter (i.e. particles), but can transfer other forms of energy (e.g. heat), to or from the other system. If, rather than an isolated system, we have a closed system, in which the entropy rather than the energy remains constant, then it follows from the first and second laws of thermodynamics that the energy of that system will drop to a minimum value at equilibrium, transferring its energy to the other system. To restate:
29:
876:
surroundings will maximize its entropy given its newly acquired energy, which is equivalent to the energy having been transferred as heat. Since the potential energy of the system is now at a minimum with no increase in the energy due to heat of either the marble or the bowl, the total energy of the system is at a minimum. This is an application of the minimum energy principle.
723:
872:
previously the potential energy of the marble, will now reside in the increased heat energy of the marble-bowl system. This will be an application of the maximum entropy principle as set forth in the principle of minimum potential energy, since due to the heating effects, the entropy has increased to the maximum value possible given the fixed energy of the system.
907:
that an external agent presses down on the weight so as to very slowly (reversibly) allow the weight to move upward to its equilibrium position, then there will be no heat generated and the entropy of the system will remain constant while energy is transferred as work to the external agent. The total energy of the system at any value of
2739:
545:
906:
is smaller than its equilibrium value. The upward force of the gas is greater than the downward force of the weight, and if allowed to freely move, the gas in the cylinder would push the weight upward rapidly, and there would be frictional forces that would convert the energy to heat. If we specify
871:
of motion of the marble. Frictional forces will convert this kinetic energy to heat, and at equilibrium, the marble will be at rest at the bottom of the bowl, and the marble and the bowl will be at a slightly higher temperature. The total energy of the marble-bowl system will be unchanged. What was
875:
If, on the other hand, the marble is lowered very slowly to the bottom of the bowl, so slowly that no heating effects occur (i.e. reversibly), then the entropy of the marble and bowl will remain constant, and the potential energy of the marble will be transferred as energy to the surroundings. The
1635:
The
Helmholtz free energy is a useful quantity when studying thermodynamic transformations in which the temperature is held constant. Although the reduction in the number of variables is a useful simplification, the main advantage comes from the fact that the Helmholtz free energy is minimized at
2070:
The minimization is with respect to the unconstrained variables. In the case of chemical reactions this is usually the number of particles or mole fractions, subject to the conservation of elements. At equilibrium, these will take on their equilibrium values, and the internal energy
518:
The first condition states that entropy is at an extremum, and the second condition states that entropy is at a maximum. Note that for the partial derivatives, all extensive parameters are assumed constant except for the variables contained in the partial derivative, but only
853:
2541:
2065:
and minimize with respect to this unconstrained variable. There may be any number of unconstrained variables depending on the number of atoms in the mixture. For systems with multiple sub-volumes, there may be additional volume constraints as well.
718:{\displaystyle \left({\frac {\partial U}{\partial X}}\right)_{S}=-\,{\frac {\left({\frac {\partial S}{\partial X}}\right)_{U}}{\left({\frac {\partial S}{\partial U}}\right)_{X}}}=-T\left({\frac {\partial S}{\partial X}}\right)_{U}=0}
2348:
2552:
141:, the internal energy will decrease and approach a minimum value at equilibrium. External parameters generally means the volume, but may include other parameters which are specified externally, such as a constant magnetic field.
1543:
1150:
The principle of minimum energy can be generalized to apply to constraints other than fixed entropy. For other constraints, other state functions with dimensions of energy will be minimized. These state functions are known as
509:
1630:
1405:
where temperature has replaced entropy as a natural variable. In order to understand the value of the thermodynamic potentials, it is necessary to view them in a different light. They may in fact be seen as (negative)
1155:. Thermodynamic potentials are at first glance just simple algebraic combinations of the energy terms in the expression for the internal energy. For a simple, multicomponent system, the internal energy may be written:
1737:
2216:
866:
Consider, for one, the familiar example of a marble on the edge of a bowl. If we consider the marble and bowl to be an isolated system, then when the marble drops, the potential energy will be converted to the
431:
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1640:
at constant temperature and volume. This follows directly from the principle of minimum energy which states that at constant entropy, the internal energy is minimized. This can be stated as:
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which varies as a system approaches equilibrium, and that it is the only such parameter which is varying. The principle of maximum entropy may then be stated as:
2734:{\displaystyle A_{0}={\underset {x}{\mathrm {min} }}({\underset {S_{0}}{\mathrm {min} }}(U(S_{0},x)-T_{0}S_{0}))={\underset {x}{\mathrm {min} }}(A_{0}(T_{0},x))}
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of the internal energy, in which certain of the extensive parameters are replaced by the derivative of internal energy with respect to that variable (i.e. the
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are the value of the internal energy and the (fixed) entropy at equilibrium. The volume and particle number variables have been replaced by
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As a concrete example of unconstrained internal variables, we might have a chemical reaction in which there are two types of particle, an
93:
2131:
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848:{\displaystyle \left({\frac {\partial ^{2}U}{\partial X^{2}}}\right)_{S}=-T\left({\frac {\partial ^{2}S}{\partial X^{2}}}\right)_{U}}
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17:
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It is seen that the energy is at an extremum at equilibrium. By similar but somewhat more lengthy argument it can be shown that
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are the respective particle numbers for these particles, then the internal constraint is that the total number of
2536:{\displaystyle A_{0}={\underset {S_{0}}{\mathrm {min} }}({\underset {x}{\mathrm {min} }}(U(S_{0},x))-T_{0}S_{0})}
50:
280:, etc.). The entropy of the system may likewise be written as a function of the other extensive parameters as
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283:
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1010:
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has been placed on top of the cylinder. It presses down on the top of the cylinder with a force of
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1138:) on the upper face of the cylinder is equal to the downward force of the mass due to gravitation (
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Alternatively, suppose we have a cylinder containing an ideal gas, with cross sectional area
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1966:
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are shown. It follows from the properties of an exact differential (see equation 8 in the
8:
2343:{\displaystyle A_{0}(T_{0})={\underset {S_{0}}{\mathrm {min} }}(U_{0}(S_{0})-T_{0}S_{0})}
273:
1538:{\displaystyle A(T,V,\{N_{j}\})={\underset {S}{\mathrm {min} }}(U(S,V,\{N_{j}\})-TS)\,}
532:
2783:
2773:
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911:
is given by the internal energy of the gas plus the potential energy of the weight:
2125:. By the definition of the Legendre transform, the Helmholtz free energy will be:
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via the equations of state. As an example of another thermodynamic potential, the
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145:
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868:
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which is greater than zero, showing that the energy is, in fact, at a minimum.
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is constant and a small change in the energy of the system would be given by:
504:{\displaystyle \left({\frac {\partial ^{2}S}{\partial X^{2}}}\right)_{U}<0}
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is the (unknown) temperature at equilibrium. Substituting the expression for
1637:
1625:{\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V,\{N_{j}\}}}
134:
1732:{\displaystyle U_{0}(S_{0})={\underset {x}{\mathrm {min} }}(U(S_{0},x))\,}
1414:
to that variable). For example, the
Helmholtz free energy may be written:
1082:=0 at equilibrium and by the principle of minimum energy, we may say that
996:
is the number of particles in the gas, and the volume has been written as
1636:
equilibrium with respect to any unconstrained internal variables for a
152:
The maximum entropy principle: For a closed system with fixed internal
2744:
showing that the
Helmholtz free energy is minimized at equilibrium.
28:
2748:
2211:{\displaystyle A(T,x)={\underset {S}{\mathrm {min} }}(U(S,x)-TS)\,}
426:{\displaystyle \left({\frac {\partial S}{\partial X}}\right)_{U}=0}
138:
1257:{\displaystyle U(S,V,\{N_{j}\})=TS-PV+\sum _{j}\mu _{j}N_{j}\,}
163:
The minimum energy principle: For a closed system with fixed
1134:
which simply states that the upward gas pressure force (
2098:
will be a function only of the chosen value of entropy
1800:
which stands for any internal unconstrained variables.
1086:=0 at equilibrium, yielding the equilibrium condition:
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2134:
2104:
2077:
2023:
1996:
1969:
1909:
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1848:
1821:
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1423:
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1280:
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53:. Unsourced material may be challenged and removed.
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2221:The Helmholtz free energy at equilibrium will be:
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1000:. Since the system is closed, the particle number
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233:
2772:(2nd ed.). New York: John Wiley & Sons.
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1078:Since the entropy is constant, we may say that
992:is pressure, μ is the chemical potential,
1552: becomes equal to the temperature since
1548:and the minimum will occur when the variable
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1267:where the intensive parameters (T, P, μ
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1042:
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113:Learn how and when to remove this message
2546:By exchanging the order of the extrema:
137:, with constant external parameters and
18:Minimum total potential energy principle
1395:{\displaystyle A(T,V,\{N_{j}\})=U-TS\,}
334:{\displaystyle S(U,X_{1},X_{2},\dots )}
234:{\displaystyle U(S,X_{1},X_{2},\dots )}
2797:
2764:
1068:{\displaystyle dU=T\,dS-PA\,dx+mg\,dx}
535:article) and from the energy/entropy
1953:{\displaystyle N_{A}=N_{1}+2N_{2}\,}
970:{\displaystyle U=TS-PAx+\mu N+mgx\,}
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129:is essentially a restatement of the
51:adding citations to reliable sources
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180:The total energy of the system is
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2017:variables with a single variable
887:. Suppose that a weight of mass
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1318:{\displaystyle (S,V,\{N_{j}\})}
156:(i.e. an isolated system), the
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2058:{\displaystyle x=N_{1}/N_{2}}
62:"Principle of minimum energy"
276:of the system (e.g. volume,
171:is minimized at equilibrium.
160:is maximized at equilibrium.
131:second law of thermodynamics
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539:that, for a closed system:
127:principle of minimum energy
10:
2821:
1124:{\displaystyle 0=-PA+mg\,}
15:
2755:, are similarly derived.
1963:we may then replace the
1153:thermodynamic potentials
1146:Thermodynamic potentials
176:Mathematical explanation
16:Not to be confused with
133:. It states that for a
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2400:{\displaystyle U_{0}}
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2373:{\displaystyle T_{0}}
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2118:{\displaystyle S_{0}}
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2091:{\displaystyle U_{0}}
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2012:
2010:{\displaystyle N_{2}}
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1983:{\displaystyle N_{1}}
1955:
1895:
1893:{\displaystyle N_{A}}
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1862:{\displaystyle N_{2}}
1837:
1835:{\displaystyle N_{1}}
1791:
1789:{\displaystyle S_{0}}
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1762:{\displaystyle U_{0}}
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1540:
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1327:Helmholtz free energy
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365:{\displaystyle X_{i}}
336:
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265:{\displaystyle X_{i}}
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735:
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443:
379:
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274:extensive parameters
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245:is entropy, and the
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47:improve this article
1408:Legendre transforms
2766:Callen, Herbert B.
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144:In contrast, for
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984:is temperature,
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146:isolated systems
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594:
593:
591:
578:
564:
556:
554:
550:
549:
547:
544:
543:
514:at equilibrium.
511:
489:
476:
472:
468:
458:
454:
453:
451:
447:
446:
444:
441:
440:
437:
433:
411:
397:
389:
387:
383:
382:
380:
377:
376:
356:
352:
350:
347:
346:
341:. Suppose that
316:
312:
303:
299:
285:
282:
281:
278:particle number
256:
252:
250:
247:
246:
216:
212:
203:
199:
185:
182:
181:
178:
119:
108:
102:
99:
56:
54:
44:
32:
21:
12:
11:
5:
2818:
2808:
2807:
2805:Thermodynamics
2793:
2792:
2778:
2760:
2757:
2742:
2741:
2730:
2727:
2724:
2721:
2716:
2712:
2708:
2703:
2699:
2695:
2690:
2686:
2683:
2680:
2674:
2671:
2668:
2663:
2659:
2653:
2649:
2645:
2642:
2639:
2636:
2631:
2627:
2623:
2620:
2617:
2610:
2606:
2601:
2598:
2595:
2589:
2584:
2580:
2577:
2574:
2568:
2563:
2559:
2544:
2543:
2532:
2527:
2523:
2517:
2513:
2509:
2506:
2503:
2500:
2497:
2492:
2488:
2484:
2481:
2478:
2473:
2469:
2466:
2463:
2457:
2450:
2446:
2441:
2438:
2435:
2429:
2424:
2420:
2394:
2390:
2367:
2363:
2351:
2350:
2339:
2334:
2330:
2324:
2320:
2316:
2313:
2308:
2304:
2300:
2295:
2291:
2287:
2280:
2276:
2271:
2268:
2265:
2259:
2256:
2251:
2247:
2243:
2238:
2234:
2219:
2218:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2171:
2167:
2164:
2161:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2112:
2108:
2085:
2081:
2052:
2048:
2043:
2037:
2033:
2029:
2026:
2004:
2000:
1977:
1973:
1961:
1960:
1946:
1942:
1938:
1935:
1930:
1926:
1922:
1917:
1913:
1887:
1883:
1856:
1852:
1829:
1825:
1811:
1802:
1783:
1779:
1756:
1752:
1740:
1739:
1727:
1724:
1721:
1718:
1713:
1709:
1705:
1702:
1699:
1694:
1690:
1687:
1684:
1678:
1675:
1670:
1666:
1662:
1657:
1653:
1633:
1632:
1619:
1614:
1610:
1606:
1603:
1600:
1595:
1589:
1586:
1581:
1578:
1572:
1567:
1564:
1546:
1545:
1533:
1530:
1527:
1524:
1521:
1518:
1513:
1509:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1479:
1475:
1472:
1469:
1463:
1460:
1457:
1452:
1448:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1403:
1402:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1367:
1363:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1314:
1311:
1306:
1302:
1298:
1295:
1292:
1289:
1286:
1283:
1268:
1265:
1264:
1250:
1246:
1240:
1236:
1230:
1226:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1193:
1189:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1147:
1144:
1132:
1131:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1076:
1075:
1064:
1061:
1057:
1054:
1051:
1048:
1045:
1041:
1038:
1035:
1032:
1029:
1025:
1022:
1019:
1016:
978:
977:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
869:kinetic energy
863:
860:
856:
855:
842:
837:
829:
825:
821:
816:
811:
807:
800:
795:
792:
789:
784:
779:
771:
767:
763:
758:
753:
749:
742:
726:
725:
714:
711:
706:
701:
695:
692:
687:
684:
678:
673:
670:
667:
660:
655:
649:
646:
641:
638:
632:
625:
620:
614:
611:
606:
603:
597:
589:
586:
581:
576:
570:
567:
562:
559:
553:
516:
515:
500:
497:
492:
487:
479:
475:
471:
466:
461:
457:
450:
422:
419:
414:
409:
403:
400:
395:
392:
386:
359:
355:
345:is one of the
330:
327:
324:
319:
315:
311:
306:
302:
298:
295:
292:
289:
272:are the other
259:
255:
230:
227:
224:
219:
215:
211:
206:
202:
198:
195:
192:
189:
177:
174:
173:
172:
161:
121:
120:
35:
33:
26:
9:
6:
4:
3:
2:
2817:
2806:
2803:
2802:
2800:
2789:
2785:
2781:
2779:0-471-86256-8
2775:
2771:
2767:
2763:
2762:
2756:
2754:
2750:
2745:
2722:
2719:
2714:
2710:
2701:
2697:
2688:
2672:
2661:
2657:
2651:
2647:
2643:
2637:
2634:
2629:
2625:
2618:
2608:
2604:
2582:
2566:
2561:
2557:
2549:
2548:
2547:
2525:
2521:
2515:
2511:
2507:
2498:
2495:
2490:
2486:
2479:
2471:
2448:
2444:
2427:
2422:
2418:
2410:
2409:
2408:
2392:
2388:
2365:
2361:
2332:
2328:
2322:
2318:
2314:
2306:
2302:
2293:
2289:
2278:
2274:
2257:
2249:
2245:
2236:
2232:
2224:
2223:
2222:
2201:
2198:
2195:
2189:
2186:
2183:
2177:
2169:
2153:
2147:
2144:
2141:
2135:
2128:
2127:
2126:
2110:
2106:
2083:
2079:
2067:
2050:
2046:
2041:
2035:
2031:
2027:
2024:
2002:
1998:
1975:
1971:
1944:
1940:
1936:
1933:
1928:
1924:
1920:
1915:
1911:
1903:
1902:
1901:
1885:
1881:
1872:
1854:
1850:
1827:
1823:
1815:molecule. If
1814:
1807:
1801:
1799:
1781:
1777:
1754:
1750:
1719:
1716:
1711:
1707:
1700:
1692:
1676:
1668:
1664:
1655:
1651:
1643:
1642:
1641:
1639:
1638:closed system
1612:
1608:
1601:
1598:
1593:
1587:
1579:
1570:
1565:
1562:
1555:
1554:
1553:
1551:
1528:
1525:
1522:
1511:
1507:
1500:
1497:
1494:
1491:
1485:
1477:
1461:
1450:
1446:
1439:
1436:
1433:
1430:
1424:
1417:
1416:
1415:
1413:
1409:
1388:
1385:
1382:
1379:
1376:
1365:
1361:
1354:
1351:
1348:
1345:
1339:
1332:
1331:
1330:
1328:
1304:
1300:
1293:
1290:
1287:
1284:
1274:
1248:
1244:
1238:
1234:
1228:
1224:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1191:
1187:
1180:
1177:
1174:
1171:
1165:
1158:
1157:
1156:
1154:
1143:
1141:
1137:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1089:
1088:
1087:
1085:
1081:
1062:
1059:
1055:
1052:
1049:
1046:
1043:
1039:
1036:
1033:
1030:
1027:
1023:
1020:
1017:
1014:
1007:
1006:
1005:
1003:
999:
995:
991:
987:
983:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
914:
913:
912:
910:
905:
902:Suppose that
900:
898:
894:
890:
886:
882:
877:
873:
870:
859:
840:
835:
827:
823:
814:
809:
798:
793:
790:
787:
782:
777:
769:
765:
756:
751:
740:
731:
730:
729:
712:
709:
704:
699:
693:
685:
676:
671:
668:
665:
658:
653:
647:
639:
630:
623:
618:
612:
604:
595:
587:
584:
579:
574:
568:
560:
551:
542:
541:
540:
538:
534:
530:
526:
522:
498:
495:
490:
485:
477:
473:
464:
459:
448:
420:
417:
412:
407:
401:
393:
384:
375:
374:
373:
357:
353:
344:
325:
322:
317:
313:
309:
304:
300:
296:
293:
287:
279:
275:
257:
253:
244:
225:
222:
217:
213:
209:
204:
200:
196:
193:
187:
170:
166:
162:
159:
155:
151:
150:
149:
147:
142:
140:
136:
135:closed system
132:
128:
117:
114:
106:
95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
42:
41:
36:This article
34:
30:
25:
24:
19:
2769:
2746:
2743:
2545:
2352:
2220:
2069:
1962:
1870:
1809:
1808:atom and an
1805:
1803:
1797:
1741:
1634:
1549:
1547:
1404:
1329:is written:
1266:
1149:
1139:
1135:
1133:
1083:
1079:
1077:
1001:
997:
993:
989:
988:is entropy,
985:
981:
979:
908:
903:
901:
896:
892:
888:
884:
880:
878:
874:
865:
857:
727:
528:
524:
520:
517:
342:
242:
179:
168:
167:, the total
164:
157:
153:
143:
126:
124:
109:
100:
90:
83:
76:
69:
57:
45:Please help
40:verification
37:
103:August 2021
2759:References
73:newspapers
2788:485487601
2644:−
2508:−
2315:−
2196:−
1585:∂
1577:∂
1523:−
1412:conjugate
1383:−
1235:μ
1225:∑
1212:−
1103:−
1034:−
949:μ
934:−
820:∂
806:∂
791:−
762:∂
748:∂
691:∂
683:∂
669:−
645:∂
637:∂
610:∂
602:∂
588:−
566:∂
558:∂
470:∂
456:∂
399:∂
391:∂
326:…
226:…
2799:Category
2768:(1985).
2749:Enthalpy
862:Examples
165:entropy
158:entropy
139:entropy
87:scholar
2786:
2776:
2353:where
1873:atoms
1742:where
980:where
895:where
512:
438:
434:
241:where
169:energy
154:energy
89:
82:
75:
68:
60:
527:, or
94:JSTOR
80:books
2784:OCLC
2774:ISBN
2751:and
2747:The
1990:and
1842:and
1769:and
998:V=Ax
496:<
436:and
125:The
66:news
1142:).
49:by
2801::
2782:.
2407::
1140:mg
1136:PA
1084:dU
1080:dS
893:mg
523:,
2790:.
2729:)
2726:)
2723:x
2720:,
2715:0
2711:T
2707:(
2702:0
2698:A
2694:(
2689:x
2685:n
2682:i
2679:m
2673:=
2670:)
2667:)
2662:0
2658:S
2652:0
2648:T
2641:)
2638:x
2635:,
2630:0
2626:S
2622:(
2619:U
2616:(
2609:0
2605:S
2600:n
2597:i
2594:m
2588:(
2583:x
2579:n
2576:i
2573:m
2567:=
2562:0
2558:A
2531:)
2526:0
2522:S
2516:0
2512:T
2505:)
2502:)
2499:x
2496:,
2491:0
2487:S
2483:(
2480:U
2477:(
2472:x
2468:n
2465:i
2462:m
2456:(
2449:0
2445:S
2440:n
2437:i
2434:m
2428:=
2423:0
2419:A
2393:0
2389:U
2366:0
2362:T
2338:)
2333:0
2329:S
2323:0
2319:T
2312:)
2307:0
2303:S
2299:(
2294:0
2290:U
2286:(
2279:0
2275:S
2270:n
2267:i
2264:m
2258:=
2255:)
2250:0
2246:T
2242:(
2237:0
2233:A
2205:)
2202:S
2199:T
2193:)
2190:x
2187:,
2184:S
2181:(
2178:U
2175:(
2170:S
2166:n
2163:i
2160:m
2154:=
2151:)
2148:x
2145:,
2142:T
2139:(
2136:A
2111:0
2107:S
2084:0
2080:U
2051:2
2047:N
2042:/
2036:1
2032:N
2028:=
2025:x
2003:2
1999:N
1976:1
1972:N
1945:2
1941:N
1937:2
1934:+
1929:1
1925:N
1921:=
1916:A
1912:N
1886:A
1882:N
1871:A
1855:2
1851:N
1828:1
1824:N
1812:2
1810:A
1806:A
1798:x
1782:0
1778:S
1755:0
1751:U
1726:)
1723:)
1720:x
1717:,
1712:0
1708:S
1704:(
1701:U
1698:(
1693:x
1689:n
1686:i
1683:m
1677:=
1674:)
1669:0
1665:S
1661:(
1656:0
1652:U
1618:}
1613:j
1609:N
1605:{
1602:,
1599:V
1594:)
1588:S
1580:U
1571:(
1566:=
1563:T
1550:T
1532:)
1529:S
1526:T
1520:)
1517:}
1512:j
1508:N
1504:{
1501:,
1498:V
1495:,
1492:S
1489:(
1486:U
1483:(
1478:S
1474:n
1471:i
1468:m
1462:=
1459:)
1456:}
1451:j
1447:N
1443:{
1440:,
1437:V
1434:,
1431:T
1428:(
1425:A
1389:S
1386:T
1380:U
1377:=
1374:)
1371:}
1366:j
1362:N
1358:{
1355:,
1352:V
1349:,
1346:T
1343:(
1340:A
1313:)
1310:}
1305:j
1301:N
1297:{
1294:,
1291:V
1288:,
1285:S
1282:(
1269:j
1249:j
1245:N
1239:j
1229:j
1221:+
1218:V
1215:P
1209:S
1206:T
1203:=
1200:)
1197:}
1192:j
1188:N
1184:{
1181:,
1178:V
1175:,
1172:S
1169:(
1166:U
1118:g
1115:m
1112:+
1109:A
1106:P
1100:=
1097:0
1063:x
1060:d
1056:g
1053:m
1050:+
1047:x
1044:d
1040:A
1037:P
1031:S
1028:d
1024:T
1021:=
1018:U
1015:d
1002:N
994:N
990:P
986:S
982:T
964:x
961:g
958:m
955:+
952:N
946:+
943:x
940:A
937:P
931:S
928:T
925:=
922:U
909:x
904:x
897:g
889:m
885:x
881:A
841:U
836:)
828:2
824:X
815:S
810:2
799:(
794:T
788:=
783:S
778:)
770:2
766:X
757:U
752:2
741:(
713:0
710:=
705:U
700:)
694:X
686:S
677:(
672:T
666:=
659:X
654:)
648:U
640:S
631:(
624:U
619:)
613:X
605:S
596:(
585:=
580:S
575:)
569:X
561:U
552:(
529:X
525:S
521:U
499:0
491:U
486:)
478:2
474:X
465:S
460:2
449:(
421:0
418:=
413:U
408:)
402:X
394:S
385:(
358:i
354:X
343:X
329:)
323:,
318:2
314:X
310:,
305:1
301:X
297:,
294:U
291:(
288:S
258:i
254:X
243:S
229:)
223:,
218:2
214:X
210:,
205:1
201:X
197:,
194:S
191:(
188:U
116:)
110:(
105:)
101:(
91:·
84:·
77:·
70:·
43:.
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
