Derive a formula, similar to equation 5.90, for the shift in the freezing temperature of a dilute solution. Assume that the solid phase is pure solvent, no solute. You should find that the shift is negative: The freezing temperature of a solution is less than that of the pure solvent. Explain in general terms why the shift should be negative.
The negative sign in the above result shows that adding a solute will lower the temperature of the freezing point of the liquid.
Here, is the chemical potential of in the solid phase and is the chemical potential of in liquid phase.
Use the equation 5.69, the chemical potential of in liquid phase is as follows.
Here, is the chemical potential of the pure solvent and is the Boltzmann's constant.
Substitute the above two equations in the equation (1) and simplify.
At the temperature , pure liquid phase is equilibrium with the solid phase. Hence,
The Gibbs free energy for pure solvent A is,
Here, is the number of particle in that phase.
Partial differentiate the equation on both sides with respect to the temperature.
Substitute for and for in the equation (2) and simplify.
Here, is the latent heat of condensation and .
Substitute for in the equation and solve for
Hence, the shift in the freezing temperature of the dilute solution is negative.
For a magnetic system held at constant TT and HH (see Problem 5.17 ), the quantity that is minimized is the magnetic analogue of the Gibbs free energy, which obeys the thermodynamic identity
Phase diagrams for two magnetic systems are shown in Figure 5.14 ; the vertical axis on each of these figures is μ0Hμ0H (a) Derive an analogue of the Clausius-Clapeyron relation for the slope of a phase boundary in the HH - TT plane. Write your equation in terms of the difference in entropy between the two phases. (b) Discuss the application of your equation to the ferromagnet phase diagram in Figure 5.14. (c) In a type-I superconductor, surface currents flow in such a way as to completely cancel the magnetic field (B, not H)(B, not H) inside. Assuming that MM is negligible when the material is in its normal (non-superconducting) state, discuss the application of your equation to the superconductor phase diagram in Figure 18.104.22.168. Which phase has the greater entropy? What happens to the difference in entropy between the phases at each end of the phase boundary?
Consider a completely miscible two-component system whose overall composition is x, at a temperature where liquid and gas phases coexist. The composition of the gas phase at this temperature is and the composition of the liquid phase is . Prove the lever rule, which says that the proportion of liquid to gas is . Interpret this rule graphically on a phase diagram.
Assume that the air you exhale is at 35°C, with a relative humidity of 90%. This air immediately mixes with environmental air at 5°C and unknown relative humidity; during the mixing, a variety of intermediate temperatures and water vapour percentages temporarily occur. If you are able to "see your breath" due to the formation of cloud droplets during this mixing, what can you conclude about the relative humidity of your environment? (Refer to the vapour pressure graph drawn in Problem 5.42.)
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