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For elements where 'liquid', is relatively easy to define, at which temperature are the most elements liquid, and which ones?

Assume 1 atm

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    $\begingroup$ Curiously, when I took science in high school (five decades ago) this would have belonged on chemistry, not physics. I majored in physics in college and we never mentioned elements except as nuclei. $\endgroup$ – Ross Millikan Feb 19 at 4:08
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    $\begingroup$ There's something that feels unsatisfying about using atmospheric pressure when discussing temperatures that wouldn't normally be associated with atmospheric pressure. I wonder if there might be a better qualifier? $\endgroup$ – Nat Feb 19 at 15:09
  • $\begingroup$ @Nat I suppose you could optimize across both temperature and pressure. $\endgroup$ – jpaugh Feb 20 at 1:20
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    $\begingroup$ chemistry.stackexchange.com/q/41971 $\endgroup$ – Mithoron Feb 20 at 19:15
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We take the natural elements (atomic numbers $1$ to $92$) to have well-defined melting and boiling points. Additionally, all figures are quoted at the standard $1\text{ atm}$ pressure. Here's some quick Python code that fetches the number of elements in the liquid state at various temperatures (note that it relies on the values of the melting and boiling points of elements defined in the mendeleev package, however it is straightforward to instead use your own dataset).

from mendeleev import element

elements = [element(i) for i in range(1, 92 + 1)] # Hydrogen to Uranium
n_liquid_list = [sum(element.melting_point < temp < element.boiling_point for element in elements) for temp in range(0, 5000)]

n_liquid_max = max(n_liquid_list)
temperature = n_liquid_list.index(n_liquid_max)

print(n_liquid_max, "elements are in the liquid state at", temperature, "K")

prints 38 elements are in the liquid state at 2161 K

For the full list, just add in

for elem in elements:
    if elem.melting_point < temperature < elem.boiling_point:
        print(elem.name, end=", ")

which gives you

Beryllium, Aluminum, Silicon, Scandium, Titanium, Vanadium, Chromium, Manganese, Iron, Cobalt, Nickel, Copper, Gallium, Germanium, Yttrium, Zirconium, Palladium, Silver, Indium, Tin, Lanthanum, Cerium, Praseodymium, Neodymium, Promethium, Gadolinium, Terbium, Dysprosium, Holmium, Erbium, Thulium, Lutetium, Platinum, Gold, Actinium, Thorium, Protactinium, Uranium

which broadly fall under transition metals, lathanides and actinides. As noted by ChrisH, there is an additional such temperature range starting at $2584\text{ K}$, in which the liquid elements are:

Beryllium, Boron, Aluminum, Silicon, Scandium, Titanium, Vanadium, Chromium, Iron, Cobalt, Nickel, Copper, Gallium, Germanium, Yttrium, Zirconium, Technetium, Ruthenium, Rhodium, Palladium, Lanthanum, Cerium, Praseodymium, Neodymium, Promethium, Gadolinium, Terbium, Dysprosium, Holmium, Erbium, Lutetium, Hafnium, Platinum, Gold, Actinium, Thorium, Protactinium, Uranium

I also thought it might be interesting to plot the total number of elements in the liquid state at a given temperature as a function of temperature:

N(Liquid) vs. Temperature

The first peak corresponds to the temperature range $2161\text{ K} - 2219 \text{ K}$ while the second peak corresponds to $2584\text{ K} - 2627 \text{ K}$.

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    $\begingroup$ The graph has 2 peaks; does this mean there are two lists? $\endgroup$ – yolo Feb 17 at 15:33
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    $\begingroup$ ... using stackoverflow.com/q/5419204 you can the 1st peak is 2161--2219K and the 2nd 2584--2627K. It's even possible that within each of these ranges the list isn't constant, if at a particular temperature one element boils and another melts. You'd need to generate the list for each temperature at which the no. of liquid elements matches the max, and compare to the previous list, outputting all unique lists with their temperatures. I suspect there will only be 2 lists though $\endgroup$ – Chris H Feb 17 at 16:06
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    $\begingroup$ The one that really stood out there is gallium - at first I thought it was a mistake that an element that melts at such a low temperature would not have boiled by then. But, indeed, it is still a liquid until 2400C. Wow. $\endgroup$ – Jon Custer Feb 17 at 16:12
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    $\begingroup$ @ChrisH great, so they coincide with my "discrete" ones then :) $\endgroup$ – Nihar Karve Feb 17 at 16:43
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    $\begingroup$ "...which gives you..." Did anyone else read the following list in a singsong voice a'la Tom Lehrer? No? Just me? $\endgroup$ – T.J. Crowder Feb 18 at 11:34
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I thought another visualization of the data set might be interesting. Here's the liquid range for each element up to and including uranium, sorted vertically from lowest to highest boiling point. (For consistency with the other answer, I also used mendeleev as the data source.) The colored vertical stripes mark the maxima found by Nihar Karve and Chris H (see the other answer).

enter image description here

This shows that the positions of the maxima are controlled by a few elements that have unusually high melting points for their boiling point: chromium and vanadium for the lower maximum, boron and ruthenium for the upper maximum.


Footnote: In mendeleev's data set, two elements have a melting point greater than their boiling point:

Name Symbol Melting point (K) Boiling point (K)
Neon Ne 48.0 27.1
Arsenic As 1090 876

I have marked these anomalies by drawing the "liquid range" line for these elements in red instead of black.

For arsenic, the anomaly is because, at 1 atm, arsenic sublimes at $876\ \mathrm{K}$. According to

Gokcen, N.A. The As (Arsenic) system. Bulletin of Alloy Phase Diagrams 10, 11–22 (1989). https://doi.org/10.1007/BF02882166

the lowest pressure at which arsenic can be liquid is $35.8 \pm 0.5\ \mathrm{atm}$ and its melting point is $1090\ \mathrm{K}$ at that pressure. The authors of mendeleev seem to have chosen to record both numbers. I presume the situation is similar for neon.

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  • $\begingroup$ Comments are not for extended discussion; this (very constructive!) conversation has been moved to chat. $\endgroup$ – rob Feb 21 at 21:39
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Elements 35 (bromine), and element 80 (mercury) are the only two elements on the periodic table that are liquid at exactly room temperature. Bromine melts at 19°F (-7.2°C), and boils at 137°F (58.8°C). Mercury melts at -37.8°F (-38.8°C), and boils at 674.1°F (356.7°C).

However, elements 31, 37, 55, and 87 (gallium, rubidium, caesium, and francium), melt slightly above room temperature. Gallium melts at 85.5°F (29.7°C), and boils at 4,352°F (2,400°C). Rubidium melts at 102.7°F (39.3°C), and boils at 1,270°F (688°C). Caesium melts at 83.3°F (28.5°C), and boils at 1,240°F (671°C). There has never been enough francium to be visibly seen before, and it has a very short half life, but its melting point is estimated to be 80°F (30°C), and its boiling point 1,300°F (630°C).

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    $\begingroup$ What do you define as "exactly room temperature"? Do you include saunas? $\endgroup$ – Chappo Hasn't Forgotten Monica Feb 19 at 4:31
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    $\begingroup$ @ChappoHasn'tForgottenMonica, if you find a sauna that's at "room temperature", it's either totally broken or not in use at the moment. (Source: I'm Finnish.) That said, room-temperature is hardly exact, but that probably doesn't matter here. $\endgroup$ – ilkkachu Feb 19 at 13:45

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