This "paradox" can be explained by carefully considering the definition of entropy. The two gases may be arbitrarily similar, but the entropy from mixing does not disappear unless they are the same gas – a paradoxical discontinuity. The additional entropy from mixing does not depend on the character of the gases it only depends on the fact that the gases are different. If the gases are the same, no additional entropy is calculated. If gas A and B are different gases, there is an entropy that arises once the gases are mixed. Again take a box with a partition in it, with gas A on one side, gas B on the other side, and both gases are at the same temperature and pressure. In this sense, it is a straightforward generalization to the argument laid out by Gibbs. The difference is that the mixing paradox deals with arbitrary distinctions in the two gases, not just distinctions in particle ordering as Gibbs had considered. The Gibbs paradox is a special case of the "mixing paradox" which contains all the salient features. This is the Sackur–Tetrode equation.Ī closely related paradox to the Gibbs paradox is the mixing paradox. Which can be easily shown to be extensive. U = 1 2 m ∑ i = 1 N ( p i x 2 + p i y 2 + p i z 2 ) The set of points in phase space that the gas could occupy is specified by the constraint that the gas will have a particular energy: This can be thought of as specifying a point in a 6 N-dimensional phase space, where each of the axes corresponds to one of the momentum or position coordinates of one of the particles. In classical mechanics, the state of an ideal gas of energy U, volume V and with N particles, each particle having mass m, is represented by specifying the momentum vector p and the position vector x for each particle. This results in the extensive Sackur–Tetrode equation for entropy, as derived next.Ĭalculating the entropy of ideal gas, and making it extensive The paradox is averted by concluding the indistinguishability (at least effective indistinguishability) of the particles in the volume. In particular, Gibbs' non-extensive entropy quantity for an ideal gas was not intended for varying numbers of particles. There is a freedom in what is defined as ordered, and it would be a mistake to conclude the entropy had not increased. If the gas particles are distinguishable, closing the doors will not return the system to its original state – many of the particles will have switched containers. Closing the door then reduces the entropy again to S per box, in supposed violation of the Second Law of Thermodynamics.Īs understood by Gibbs, and reemphasized more recently, this is a misapplication of Gibbs' non-extensive entropy quantity. In fact, the non-extensive entropy quantity defined and studied by Gibbs would predict additional entropy. The entropy of the gas in the two-container system can be easily calculated, but if the equation is not extensive, the entropy would not be 2 S. No macroscopic changes occur, as the system is in equilibrium. Now a door in the container wall is opened to allow the gas particles to mix between the containers. There is a certain entropy S associated with each container which depends on the volume of each container. in volume, mass, temperature, pressure, etc). The gas in container #1 is identical in every respect to the gas in container #2 (i.e. Two identical containers of an ideal gas sit side-by-side. Gibbs himself considered the following problem that arises if the ideal gas entropy is not extensive. If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, in the thermodynamic limit, the paradox is averted. A related paradox is the " mixing paradox". The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. This leads to a paradox known as the Gibbs paradox, after Josiah Willard Gibbs, who proposed this thought experiment in 1874‒1875. In statistical mechanics, a semi-classical derivation of entropy that does not take into account the indistinguishability of particles yields an expression for entropy which is not extensive (is not proportional to the amount of substance in question). From a consideration of the energies involved $w$ is expected to be positive.Not to be confused with Gibbs phenomenon. I want to derive an expression for the internal energy of mixing, $\Delta_\mathrm$ all of which are negative.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |