lecture18 - Superheated and supercooled materials...

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Unformatted text preview: Superheated and supercooled materials (continued from the last lecture). In the last lecture we have explained why gases can be supercooled below their condensation point: it has to do with the energetic penalty to create a liquid droplet, which comes from the energy cost to create a liquid ­gas interface. In practice, condensation can often be initiated on the walls of the container, which avoids this energetic penalty. Superheated liquid is somewhat easier to create (even in your microwave) – it will be stable with respect to the formation of small vapor bubbles; again, impurities will favor vaporization. Supercooled vapor and superheated liquid states are essential for the operation of, respectively, cloud (Wilson) chambers and bubble chambers, which are devices use to observe the trajectories of elementary particles. Supercooled liquid is another common phenomenon. Superheated solids are very hard to create as they will tend to melt at their surfaces. Diamond is an example of a supercooled carbon (it should be graphite at room temperature). An example of a supercooled liquid: Honey flows even if put in a fridge. It can be crystallized if impurities are added. What happens if I continue to lower temperature of a supercooled a liquid? As long as it does not have enough time to crystallize, it can end up in state of “frozen disorder”, or glass. In this state, the liquid no longer flows, at least for any practical purpose. The exact nature of this glassy state is still not fully understood and remains somewhat controversial. Consider, for example, what happens to the entropy of the liquid as it is being supercooled down past the freezing point. We know that the entropy of any material, at a constant P, changes according to: S (T ) = S (Tm ) + ∫ dTC p / T Tm T The heat capacity of a liquid is generally greater than that of a solid, thus S(T) of a liquid will go down faster than that of a solid. This implies that at some point Sl(T) would become lower than that of the crystalline solid! This doesn’t make sense as the solid is more ordered than liquid. This observation is often referred to as “entropy crisis” or Kauzmann’s paradox (Kauzmann was the first to point this out). There are several alternative explanations of this paradox. One proposal is that there is transition to a new “glass” phase, which is fundamentally different from that of a liquid. Resolution of this paradox is still an open question. Thermodynamics of mixtures The chemical potential revisited. For a mixture that consists of n1 moles of compound 1, n2 moles of compound 2 etc…, the Gibbs free energy is a function of P, T, n1, n2, … G = G(P, T, n1, n2, …) We can write dG = ⎜ If we keep all ni’s constant, the mixture behaves like a single ­component system and therefore ⎛ ∂G ⎞ ⎛ ∂G ⎞ ⎛ ∂G ⎞ dn1 + ... ⎟ ⎟ dP + ⎜ ⎟ dT + ⎜ ∂n1 ⎠ P,T ,n ,... ⎝ ∂P ⎠T ,n ⎝ ∂T ⎠ P,n ⎝ 2 " !G % = V $ !P ' # & T ,{n } i " !G % = ( S $ !T ' # & P ,{n } i Let us introduce the notation: " !G % $ !n ' # i & P ,T ,n = µi ( P , T , n ) j , j (i ,... This is the chemical potential of component i in a multicomponent system. We have then: dG = VdP ­SdT + ∑ µ dn , i i (1) which is more general than the expression we previously had. Now suppose I have a container with a mixture of n1 moles of material 1, n2 moles of material 2 etc. If I double the number of moles of each compound (at the same P and T), I will not change the physical properties of the mixture and so the Gibbs energy, which is an extensive quantity) will also be doubled. G(P, T, 2n1, 2n2, …) = 2 G(P, T, n1, n2, …) I can generalize this expression and write: G(P, T, an1, an2, …) = a G(P, T, n1, n2, …) If we differentiate this expression with respect to a, we will get: dG(P, T, an1, an2, …)/da = " !G ( P , T , an1 , ...) % $ ' !(an1 ) # &n n2, …) j , j (1, P ,T d (an1 ) " !G ( P , T , an1 , ...) % +$ ' da !(an2 ) # &n j , j ( 2 , P ,T d (an2 ) + ... = G(P, T, n1, da This equation is valid for any a including a=1. Setting a=1, we find: G(P, T, n1, n2, …) = ∑ µ n ii (2) You may notice that we have already seen this expression in the case of single component systems. Note, however, that the chemical potentials are, generally speaking, functions of n1, n2 and are different from those of pure compounds. ...
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