This problem is XXXXXXXXXXfrom a book "Thermodynamics and Statistical Mechanics An Integrated...

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This problem is XXXXXXXXXXfrom a book  "Thermodynamics and Statistical Mechanics An Integrated Approach by M. Scott Shell"Extracted text: 6.20. If placed in very smooth, clean containers, many liquids can be supercooled to
temperatures below their melting points without freezing, such that they remain
in the liquid state. If left long enough, these systems will eventually attain equilib-
rium by spontaneously freezing. However, in the absence of nucleation sites like
dirt or imperfections in the walls of the container, the time required for freezing
can be long enough to permit meaningful investigations of the supercooled state.
Consider the spontaneous freezing of supercooled liquid water. For this prob
lem you will need the following data taken at atmospheric pressure: cp 38 J/K
mol for ice (at 0 °C, but relatively constant over a small T range); Cp-75 J/K mol
for liquid water (at 0 °C, but relatively constant); and Δhm-6, 026 J/mol for liquid-
ice phase equilibrium at 0 °C. Here Ax indicates iquid Xcrystal for any property x
Recall that E + PV, h = H/N, and Cp = (ah/01),-7(os/OT)P, and that h and s
are state functions
(a) In an isolated system, when freezing finally occurs, does the entropy increase
or decrease? What does this imply in terms of the relative numbers of micro-
states before and after freezing? Is "disorder" a useful qualitative description of
entropy here?
(b) If freezing happens fast enough, to a good approximation the process can be
considered adiabatic. If the entire process also occurs at constant pressure,
how that freezing is a constant-enthalpy (isenthalpic) process
(C) If one mole of
supercooled water is initially at -10 °C (the temperature of a
typical household freezer), what fraction freezes in an adiabatic process as it
comes to equilibrium? Compute the net entropy change associated with this
event. Note that this is not a quasi-static process.
(d) Consider instead the case when the freezing happens at constant T and P.
Under these conditions, explain why the expression Δ:-Δh/T is valid when
liquid water freezes at its melting point, but not when supercooled liquid
water freezes.
(e) Compute ASwater, ASsurw and ASworld when one mole of supercooled liquid
ater spontaneously freezes at -10 °C (e.g, in the freezer). Assume that the
surroundings act as a bath

Aditi · Accepted Answer

ANSWER
a. For an isolated system, dH = TdS. Thus,
Freezing of supercooled water implies that, T ≤ 273.15 K.
As T  0, thus the process becomes irreversible. This happens when heat flows out of the system and freezing occurs, increasing the entropy of the system.
Supercooled liquids, rather than ice, develop at temperatures of 0 °C (273.15 °F) in a closed system. It is necessary to remove heat from the system in order for ice to develop at this temperature, but because the system is isolated, this is no longer possible. When the supercooled liquid is jolted (disordered), it forms ice in a fraction of a second. The entropy principle predicts that disorder will increase.
b.

6.20. If placed in very smooth, clean containers, many liquids can be supercooled to temperatures below their melting points without freezing, such that they remain in the liquid state. If left long...

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