Can we compare the vapour pressure of solutions whose concentrations and temperature is known?

Question

If we have some solutions, in which we know the temperature and concentration of the solute particles, can we compare their vapour pressures?

Original question:

Which solution has the highest vapour pressure among the following?

a) $0.02 ~\pu M ~\ce{NaCl}$ at $50^{\circ}$C b) $0.03 ~\pu{M}$ sucrose at $15^{\circ}$C c) $0.005 ~\pu{M} ~\ce{CaCl_2}$ at $50^{\circ}$C d)$0.005~ \pu{M} ~ \ce{CaCl_2}$ at $25^{\circ}$C

I answered this question by noting that at higher temperature and lower concentration, vapour pressure of solution should be the highest because more temperature implies more kinetic energy which means more volatility and low concentration means that the solute is less solvated so it forms less bonds with the solvent and therefore has more vapour pressure, so answer is c). Is this reasoning correct?

Now I think that these options were set so that it becomes easy to compare. But what if I had options like $0.02 ~\pu M ~\ce{NaCl}$ at $50^{\circ}$C ,$0.03 ~\pu M~ \ce{NaCl}$ at $60^{\circ}$C, $0.005 ~\pu M ~\ce{CaCl_2}$ at $20^{\circ}$ C, how shall we compare their vapour pressures without doing experiments?

Now, if there existed a relation between vapour pressure and concentratin or temperature, it would become easier.

For example, relation between vapour pressure and temperature can be given by Antoine equation, but I want something that also incorporates the concentration in the equation also, so that by plugging in the values, it is easy to compare.

I am also not sure, whether van't hoff factor will have a role in it or not.

Answer 1

The point of the question is to test your understanding

i. of Raoult's law (that increasing the solute concentration decreases the vapor pressure and that the identity of the solute is not important in an ideal solution; the mole fraction of solute determines $\Delta p$: $\Delta p = p-p^\circ = -\chi_\textrm{solute}p^\circ$

ii. that vapor pressure increases with T

Other things you might want to remember are that molality and molarity are linearly proportional for dilute solutions (ie then $m \propto c_M$), and that ideality is approached for dilute solutions of small solutes (typically at low $\pu{mM}$ concentrations such as in this problem).

You can answer the question based entirely on consideration of the van't Hoff factor to compute total concentrations, and by assuming that the vapor pressure increases with T.

First compute effective total solute concentrations $im$ (equal to either the solute concentration for a non-dissociating solute such as sucrose, or the total ion concentration for the electrolyte solutions):

$$\begin{array}{|c|c|c|} \hline \text{species} &T (\pu{^\circ C})& c_M (\pu{M})& i & ic_M(\pu{M}) \\\hline 1.~\ce{NaCl} & 50 & 0.02 & 2 & 0.04\\ 2.~\text{sucrose} & 15 & 0.02 & 1 & 0.02\\ 3.~\ce{CaCl2 } & 15 & 0.005 & 3 & 0.015\\ 4.~\ce{CaCl2 } & 50 & 0.005 & 3 & 0.015\\\hline \end{array}$$

Inspection should quickly eliminate all options but row 4 since they have a higher effective concentration at the same T, or since they are at a lower T at the same effective concentration.

Actual vapor pressures are not stated in the problem. In the absence of that information there is no point in attempting to compare things in more detail, and that's not the point of the problem. However, provided Raoult's law holds it can be used to determine the change in vapor pressure regardless of T. Raoult's law assumes the identity of the solute does not matter, that the effect of added solute on the activity of solvent is entirely entropic (accounted for by ideal mixing entropy) and that the gas is ideal. In general, to predict vapor pressure at any T and solute concentration you need a model of the activity of the solvent as a function of the solute concentration, and of the activity in the gas phase. You can determine the vapor pressure by finding the conditions that render the activities identical.