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For example, I have two beakers each containing water. One is at 90 $^{\circ}$C and one is at 10 ${^\circ}$C.

Will the final temperature of the solution I make from mixing the two components be different if I slowly poured the 90 ${^\circ}$C water into the 10 ${^\circ}$C water and vice versa?

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    $\begingroup$ No - the temperature of the final solution is not dependent on the order in which the components are mixed together. $\endgroup$
    – Todd Minehardt
    May 19, 2016 at 17:35
  • $\begingroup$ If that's the case I would appreciate it as an answer. Also, not sure why I have a downvote. $\endgroup$
    – Terry
    May 19, 2016 at 17:51
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    $\begingroup$ If the components may chemically react together, the answer is surely yes. I'm sure any reasonably competent chemist could even come up with examples where mixing the components together in one order is perfectly harmless, while mixing them in another order will produce a violent explosion. $\endgroup$
    – Ilmari Karonen
    May 19, 2016 at 20:47

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Temperature is a so-called state function:

In thermodynamics, a state function or function of state is a function defined for a system relating several state variables or state quantities that depends only on the current equilibrium state of the system. State functions do not depend on the path by which the system arrived at its present state. A state function describes the equilibrium state of a system.

The final temperature of your system does not depend on how the system got there (order of mixing, for example).

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Whilst the order of mixing does not matter in theory (see e.g. Todd's answer) it can play a role in praxis: Consider an imperfectly isolated system (e.g. a coffe cup) in air. The loss of heat is among other things proportional to the temperature gradient (basically local difference in temperature). An intuitive example would be that it matters whether you let your tea cool for a while and then add milk, or add milk and then let it cool. The former will be colder after the same time.

Another thing to consider is that it gets more complicated once you have changes in phase. E.g. a molten salt will likely have much more energy than a solution at the same temperature. If you heat the components separately and later mix you will have a higher temperature than when you first make the solution and then heat them. (The example with salt might be a bit contrived as they generally have high melting points.)

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