Thermodynamics and Coffee

Question

Heyo, I'm sure you all know of the experiment where you take a mixture of ice and water, plop some salt in there and the temperature of the solution progressively drops. The salt increases the entropy of the liquid state, shifting somewhat the ice/water equilibrium towards water. As ice is converted to liquid water, the endothermicity of this process leads to a decrease in the temperature of the solution.

Now, I was wondering if the reverse would ever be possible, say you were to have a thermos flask which would be sealed. It would contain Hot Tea and have a slot where one could pour in salt grains without releasing steam. By the logic in the previous question, won't some of the water vapor move to the liquid state (exothermic), resulting in the heatin up of your already hot tea. If possible would this change even be measurable or useful in any sort of way?

Answer 1

You're exactly right: the phenomenon is called boiling-point elevation:

Boiling-point elevation describes the phenomenon that the boiling point of a liquid (a solvent) will be higher when another compound is added, meaning that a solution has a higher boiling point than a pure solvent. [...] It is an effect of the dilution of the solvent in the presence of a solute. It is a phenomenon that happens for all solutes in all solutions, even in ideal solutions, and does not depend on any specific solute-solvent interactions. The boiling point elevation happens both when the solute is an electrolyte, such as various salts, and a nonelectrolyte. In thermodynamic terms, the origin of the boiling point elevation is entropy and can be explained in terms of the vapor pressure or chemical potential of the solvent. In both cases, the explanation depends on the fact that many solutes are only present in the liquid phase and do not enter into the gas phase (except at extremely high temperatures).

The empirical method for computing boiling point elevation for a given solute is described here. In short:

$\Delta T_b = \sum_i{n_iK_{b,i}m_i}$

Here, $\Delta T_b$ is the observed increase in boiling point, and $K_{b,i}$ and $m_i$ are the boiling point elevation constant and the molality (moles of solute per kilogram of solvent), respectively, of solute $i$. The $n_i$ factor is the number of ions into which solute $i$ dissociates upon dissolution (e.g., $n_{sucrose} = 1$ and $n_{\ce{NaCl}} = 2$).

Your best bet for a full thermodynamic description is a college or graduate level textbook on the topic. I don't know offhand of any that specifically address boiling-point elevation, though. The Wikipedia page on colligative properties actually has a bit of thermodynamic detail, providing the expression for computing the boiling point elevation constant from the enthalpy of vaporization $(\Delta H_{\mathrm{vap}})$ and other physical parameters:

$K_{b,i} = \frac{RM_iT_b^2}{\Delta H_{\mathrm{vap}}}$

where $R$, $M$, and $T_b$ are the ideal gas constant, the formula weight of solute $i$, and the boiling point of pure solvent, respectively.