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This question is out of pure curiosity.

At what temperature are a majority of the elements on the periodic table in a liquid state/phase of matter? For the purpose of this question, assume the standard pressure (100,000 Pa). Is there some sort of formula or theorem that explains this? Please give a detailed explanation on your conclusion.

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    $\begingroup$ The Dynamic Periodic Table is a great site for questions about the properties of different elements. For your purposes, it even has a slider for the phases of different elements at different temperatures. I will say though that carbon is the last element to melt, and at atmospheric pressures it doesn't even do that-- it actually sublimes at $3,900\:\mathrm{K}$. $\endgroup$ – ringo Dec 10 '15 at 8:06
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Now that's a great question indeed! Evidently, at 0K all elements except helium are solids, at 10000K they are all gases, so someplace in between the number of liquids must reach a maximum; what and where might that be?

Well, there is no formula or theorem that says liquid hydrogen must boil at 20K, nor is there such a thing for any other element, so this line of thinking hardly provides any insight. Let's try another way.

Let's take a look at all elements' melting and boiling points. (I used the great site referenced in a comment by ringo, where all data are conveniently stored on line 47 in the file http://www.ptable.com/Static/interactivity-2fcd37b.js). Let's simply make a joint list of all melting and boiling points, sort it (keeping track of which is which), and then walk through it from bottom to top, calculating the number of liquids on each interval. If you know some kind of programming, you can do it with a script.

It turns out that the maximum number of simultaneously liquid elements is 49, and it is reached several times between 2183K (melting of vanadium) and 2792K (boiling of aluminium). By dragging the slider you may actually see which elements are included in these 49. Now, the data are not terribly reliable (for example, in higher actinides they claim some melting points obtained by God-knows-what extrapolation, but no boiling points at all), yet still this is something to think about.

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The boiling point of hydrogen is $-252.87~^{\circ}\rm C$ or $20.28\rm\,K$, which is the least boiling point so you can expect that all other elements would liquidify or solidify at or below this temperature.

But the concept of phases is much more complicated than that. Aside from three states (namely solid, liquid and gas) there are other states such as plasma, fermionic condensate, Bose–Einstein condensate and top-quark condensate.

At Bose–einstein condensate, at 170 nano-kelvin rubidium atoms coalesce to form one particle that acts like a wave forming Bose-gas. So, from this example you can perceive that on super-cooling you can form gas. Hence, it is very difficult to specifically answer your question as technology improves nature tend to show its diverse beauty.

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