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Lets say supercooled liquid water at $263\ \mathrm K$ isobarically changes to solid ice at the same temperature. I wish to calculate the change in entropy of the surroundings and I happen to know the $\Delta H$ for the reaction.

This should be simple enough. Since the process is isobaric, $q_\text{rev} = \Delta H$ and $q_\text{surr}=-q_\text{reac}$

$\Delta S_\text{surr} = \frac{-\Delta H}{T} $

But what should T be here? Is it $263\ \mathrm K$? Or the temperature of the surroundings? Or are they both the same, ie;

Do we consider the temperature at which the reaction is taking place to be the temperature of the surroundings as well?

If so, why?

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  • $\begingroup$ The phase transition no reaction. Yes. That would be the minum for that process $\endgroup$ – Alchimista Jan 19 at 9:41
  • $\begingroup$ Correction: Unless its an adiabatic process the surroundings have to be at the same T as the system (isothermal), otherwise its not at equilibrium. $\endgroup$ – Try Hard Jan 19 at 12:53
  • $\begingroup$ In thermodynamics, the surroundings are typically treated as an ideal constant temperature reservoir whose temperature T is specified. And for such a reservoir, its entropy change is always $q_{surr}/T$, irrespective of any irreversibilities in the system. Also, the temperature at the interface between the system and the surroundings is always the reservoir temperature, not the average temperature of the system. $\endgroup$ – Chester Miller Jan 19 at 13:30
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Thermodynamics deals with changes between systems at equilibrium, which is to say that the initial and final states of a process are regarded as equilibrium states. As explained in the comments, the surroundings are usually defined as an ideal reservoir of infinite size and thus infinite heat capacity and constant temperature, clearly an approximation, but the key point remains that stated in the previous sentence.

It does not make sense to define an equilibrium diathermal state in which the surroundings and the system have a different temperature. Since in the stated problem the system undergoes exchange of heat with its surroundings, and the system is at an initial and final T of 263 K after the transition, then the surroundings have to be at 263 K throughout.

One potential point of confusion regarding this type of problem concerns the metastable nature of the supercooled state. This is not a true thermodynamic equilibrium but is regarded as sufficiently stable to be regarded as such. An additional point of confusion may be that heat is transferred to the surroundings even though system and surroundings are at the same T. This is possible because the surroundings have infinite size and heat capacity. One way to envision such surroundings is as a reservoir of a substance with a melting point of 263 K.

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