how does the volume/mass combination of a liquid at two different temperatures resolve to the third temperature when they are combined?

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The molecules from the two different parts mix and collide with one another. Higher-energy (warmer) particles transfer some heat energy to lower-energy (cooler) particles when they collide. Eventually, the heat energy is distributed equally.

Imagine you have two cups of juice, one with a little bit of juice and one with a lot of juice. If you pour the two cups of juice together, you’ll have more juice in the end because you combined the juice from both cups.

Now let’s say one cup of juice is cold and the other cup is hot. When you mix them together, the hot juice will warm up the cold juice and make it hotter, and the cold juice will cool down the hot juice and make it cooler.

The amount of cooling or heating depends on how much juice there is and how hot or cold it is to begin with. But if you know the volume (how much juice there is) and the mass (how heavy the juice is) of each cup of juice, you can figure out what temperature the mixed juice will be.

It’s kind of like a recipe. If you know how much of each ingredient you have and how hot or cold they are, you can figure out what the final product will be like. In this case, the final product is the mixed juice at a new temperature.

What we perceive as temperature is simply a measure of the average kinetic energy of the substance (the energy of the molecules in the bulk is what’s called Normally Distributed). When we mix substances at different temperatures they quickly exchange energy to get rid of the temperature difference between them so the average energies combine rapidly. Once this happens, the molecules arrange themselves according to the energy they have (the density at that specific temperature).

Mass is conserved so the mass of the mixture is simply the sum of the masses of the two liquids being mixed; volume can be found by dividing the total mass by the density (or by measuring it).

It’s just a simple weighted average of the two.

Mass1 * Temp1 + Mass2 * Temp2 divided by the total mass.