From the Gibbs-Duhem equation for a pure species, 0 = S d[partial differential U
ID: 478297 • Letter: F
Question
From the Gibbs-Duhem equation for a pure species, 0 = S d[partial differential U/partial differential S)_V, N] + V d[partial differential U/partial differential V)_S, N] + N d[(partial differential U/partial differential N)_S, V] show that at a fixed temperature, the derivative of the chemical potential of a pure species with respect to pressure is equal to the molar volume of that species, and at a fixed pressure, the derivative of the chemical potential of a pure species with respect to temperature is equal to the negative of the molar entropy of that species. Compare the above results with your experience of how concentration depends on temperature (at fixed pressure) and pressure (at fixed temperature). How do signs of these concentration derivatives compare with the signs of the above chemical potential derivatives? How does this suggest that the chemical potential varies with concentration?Explanation / Answer
(a) For a one component system, ‘molar Gibbs energy’ and ‘chemical potential’ are synonyms, so = Gm.
we know that dG = Vdp SdT
implies
Since Gm =
(i)
(ii)
(b)
At fixed pressure, the concentration will decrease with increase in temperature. From (i) the chemical potential also decreases with increase in temperature at constant pressure. This will lead to the conclusion that if chemical potential is more then concentration of that substance is more.
At fixed temperature, the concentration will increase with increase in pressure. (Example: increase in solubility of carbondioxide in soda with increase in pressure) From (ii) the chemical potential also increases with increase in pressure at constant temperature. This will again lead to the same conclusion that if chemical potential is more then concentration of that substance is more
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