PART IIIPHYSICAL INTERPRETATIONSSECTION AOF THE SIMPLE REVERSIBLE OPERATIONS IN THERMODYNAMICSChange under Constant VolumeWe found above that the entropy of a state was precisely defined in a physical way by the number of complexions of that state. Now let us see what happens to this number of complexions when an ideal gas experiences some of the simpler changes, of a reversible (non-cyclical) character. We will begin with the case in which the volume of the gas remains constant while its temperature rises, the final state of the gas having a higher temperature than its initial state.We see from Eq. (7),p. 51, thatgrows and from Eq. (4),p. 50, thatdiminishes. MAXWELL'S Law, given by Eq. (5),p. 50, shows for a given velocitythat the numberof molecules possessing the given velocity islessin the final state than it was in the initial state, and as the total numberof molecules in the gas is unchanged, there will be a greater variety of velocities in the final state. This makes the number of possible permutations greater in the final state, thus increasing the number of complexions; consequently, as entropy varies with the logarithm of the number of complexions, we see that the entropy of the final state is greater than in the initial state and this agrees with experience.Isobaric ChangeNext we interpret how the number of complexions are affected by isobaric change during a reversible process, again assuming that the temperature in the final state is greater than in the initial one. Here the steps and the conclusion are exactly the same as in the preceding case. In both cases just the opposite result is reached when there is afallin temperature.As thediagram contains the co-ordinates, and represents mainly the mechanical changes in the body under consideration, we can, by suitable combination, similarly interpret any other reversible change of state represented in thisdiagram.Isothermal ChangeHowever, because of its general importance and because of its bearing on the temperature-entropy diagram, we will here also tell, in the same physical terms, what happens when our ideal gas undergoes isothermal change with increase of volume. As the temperature in the final state is equal to that in the initial one, the quantitydoes not change and thereforedoes not change nor (see Eq. (5),p. 50) does the numberof molecules possessing the velocitychange. The variety of velocities in the final state is therefore the same as in the initial state and does not at all contribute to that necessary increase in the number of complexions (configurations) for which we are looking.Thedirectionof the velocity of a molecule would be another variety element, but as the final volume evidently possesses as many velocity directions as the initial volume, this element or co-ordinate will not contribute to increased complexity in the final state. But, as the volume has increased, the final state will contain more unit volumes (and these can be taken as smallas we please) than the initial state. As it is here equally likely that a particular molecule will be found in any one of these unit volumes, it is evident that the increase of volume will add increased variety to the location or configuration of the molecules and by indulging in the swapping of places inherent in the production of complexions, we see that said increment of volume will make the number of complexions in the final state greater than in the initial state, which in turn means that the entropy in the final state is also greater. This accords with experience, but it can also be seen from the formuladerived by PLANCK (p. 63 of Vorl. ü. Theor. Physik), from probability considerations, for the state of thermal equilibrium. Hereis the universal constant (seep. 66) and the other terms have the same meaning as before.Isentropic ChangeThe last reversible process, to be here physically interpreted, is isentropic change from the initial state of thermal equilibrium to its final state. Evidently only the physical elements underlying the bracketed term in Eq. (31) need to be considered.As we are considering isentropic change (), it does not make any difference whether on the one hand we think of this isentropic change as accompanied by an increase in temperature and decrease in volume, or on the other hand think of said change as taking place with decrease of temperature and increase of volume. Suppose we assume the latter kind of change. Then from what has preceded we know that increase of volume by itself would increase the number of complexions of the final state, also, from what has gone before, we know that the drop in temperature by itself will lead to decrease in the number of complexions in the final state. These two influences actingsimultaneously therefore tend to neutralize each other and if they exist in the proper ratio, derivable from the bracketed quantity in Eq. (31), they will completely balance each other and produce no change whatever in the number of complexions while passing from the initial to the final state of equilibrium, i.e., will produce no change whatever in the entropy of the gas under consideration. In isentropic change Nature has no preference for its various states.The temperature-entropy diagram considers mainly thermal changes, and as we have considered the influence of both of its co-ordinates in the number of complexions, we can ascertain by proper combination, foranyreversible change of state, the corresponding character of the change in the number of complexions. It is evident, too, that in the diagram any reversible change of state is equivalent, so far as the change of entropy in the one body is concerned, to an isentropic change combined with an isothermal change, the latter only affecting the result, so far as change in number of complexions is concerned.
Change under Constant Volume
We found above that the entropy of a state was precisely defined in a physical way by the number of complexions of that state. Now let us see what happens to this number of complexions when an ideal gas experiences some of the simpler changes, of a reversible (non-cyclical) character. We will begin with the case in which the volume of the gas remains constant while its temperature rises, the final state of the gas having a higher temperature than its initial state.
We see from Eq. (7),p. 51, thatgrows and from Eq. (4),p. 50, thatdiminishes. MAXWELL'S Law, given by Eq. (5),p. 50, shows for a given velocitythat the numberof molecules possessing the given velocity islessin the final state than it was in the initial state, and as the total numberof molecules in the gas is unchanged, there will be a greater variety of velocities in the final state. This makes the number of possible permutations greater in the final state, thus increasing the number of complexions; consequently, as entropy varies with the logarithm of the number of complexions, we see that the entropy of the final state is greater than in the initial state and this agrees with experience.
Isobaric Change
Next we interpret how the number of complexions are affected by isobaric change during a reversible process, again assuming that the temperature in the final state is greater than in the initial one. Here the steps and the conclusion are exactly the same as in the preceding case. In both cases just the opposite result is reached when there is afallin temperature.
As thediagram contains the co-ordinates, and represents mainly the mechanical changes in the body under consideration, we can, by suitable combination, similarly interpret any other reversible change of state represented in thisdiagram.
Isothermal Change
However, because of its general importance and because of its bearing on the temperature-entropy diagram, we will here also tell, in the same physical terms, what happens when our ideal gas undergoes isothermal change with increase of volume. As the temperature in the final state is equal to that in the initial one, the quantitydoes not change and thereforedoes not change nor (see Eq. (5),p. 50) does the numberof molecules possessing the velocitychange. The variety of velocities in the final state is therefore the same as in the initial state and does not at all contribute to that necessary increase in the number of complexions (configurations) for which we are looking.
Thedirectionof the velocity of a molecule would be another variety element, but as the final volume evidently possesses as many velocity directions as the initial volume, this element or co-ordinate will not contribute to increased complexity in the final state. But, as the volume has increased, the final state will contain more unit volumes (and these can be taken as smallas we please) than the initial state. As it is here equally likely that a particular molecule will be found in any one of these unit volumes, it is evident that the increase of volume will add increased variety to the location or configuration of the molecules and by indulging in the swapping of places inherent in the production of complexions, we see that said increment of volume will make the number of complexions in the final state greater than in the initial state, which in turn means that the entropy in the final state is also greater. This accords with experience, but it can also be seen from the formuladerived by PLANCK (p. 63 of Vorl. ü. Theor. Physik), from probability considerations, for the state of thermal equilibrium. Hereis the universal constant (seep. 66) and the other terms have the same meaning as before.
Isentropic Change
The last reversible process, to be here physically interpreted, is isentropic change from the initial state of thermal equilibrium to its final state. Evidently only the physical elements underlying the bracketed term in Eq. (31) need to be considered.
As we are considering isentropic change (), it does not make any difference whether on the one hand we think of this isentropic change as accompanied by an increase in temperature and decrease in volume, or on the other hand think of said change as taking place with decrease of temperature and increase of volume. Suppose we assume the latter kind of change. Then from what has preceded we know that increase of volume by itself would increase the number of complexions of the final state, also, from what has gone before, we know that the drop in temperature by itself will lead to decrease in the number of complexions in the final state. These two influences actingsimultaneously therefore tend to neutralize each other and if they exist in the proper ratio, derivable from the bracketed quantity in Eq. (31), they will completely balance each other and produce no change whatever in the number of complexions while passing from the initial to the final state of equilibrium, i.e., will produce no change whatever in the entropy of the gas under consideration. In isentropic change Nature has no preference for its various states.
The temperature-entropy diagram considers mainly thermal changes, and as we have considered the influence of both of its co-ordinates in the number of complexions, we can ascertain by proper combination, foranyreversible change of state, the corresponding character of the change in the number of complexions. It is evident, too, that in the diagram any reversible change of state is equivalent, so far as the change of entropy in the one body is concerned, to an isentropic change combined with an isothermal change, the latter only affecting the result, so far as change in number of complexions is concerned.