Change of temperature upon dilution Consider a concentrated dye solution, say 10µM of Rhodamine 6G in water, which is rapidly diluted down to 10nM in an open container. I'm trying to estimate the (presumably minute) temperature change, assuming both solutions (dye and water) are initially at the same room temperature. I believe dilution increases entropy ever so slightly, but I'm struggling to turn this statement into a quantitative temperature change. Specifically, how do I calculate the change in entropy? Once I know this, the change in temperature should be straight-forward.
At such low concentrations, can’t you just assume an ideal solution? Then the  change in entropy from mixing would be $\Delta S = -R(x_A\ln x_A + x_B \ln x_B)$. But because the solution is so dilute, you can see that both terms are going to be essentially zero. (For a 10 nM solution, the contribution from entropy of mixing is on the order of $10^{-8}$ J K$^{-1}$ mol$^{-1}$.)

The dilution process would thus lower the total entropy, since there’s less uncertainty in the system. (You’d get the value you compute at 10 nM, the final state, minus the value at 10 µM, the initial state.)