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I have a quick simple question that I can't seem to find an answer for.

Say my optimal condition for something to happen -- doesn't matter what -- happens at Pressure: 10$^{-10}$ Torr and 600 K.

Now I want to know what temperature I must set if I increase my pressure to 10$^{-6}$ Torr to recreate the condition.

What formula should I use?

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    $\begingroup$ It is completely unclear what your "optimal condition" is. Please provide a concrete example. $\endgroup$ – Sanya Aug 15 '16 at 11:22
  • $\begingroup$ Growth of a crystal structure on a substrate. $\endgroup$ – sci-guy Aug 15 '16 at 11:35
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    $\begingroup$ That is a very complex problem depending on the crystallising substance and the surface. It is weird to expect a good answer if you let yourself be asked for every single detail. $\endgroup$ – Sanya Aug 15 '16 at 11:39
  • $\begingroup$ Hmm ok, I thought there was a simple relationship. $\endgroup$ – sci-guy Aug 15 '16 at 11:52
  • $\begingroup$ Note : from $10^{-10}$ torr to $10^{-6}$ torr is an increase in pressure by a factor of 10,000 - not a decrease. $\endgroup$ – sammy gerbil Aug 15 '16 at 12:06
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Although as others have mentioned your question is rather ambiguous since the "something" you want to happen in a condition should be clarified for a better answer.

But in the context of "Proportional Reasoning" there is a general rule of thumb that you can follow:

First make sure that the occurrence of that "something" depends solely on pressure and temperature and nothing else then form an equality of below form:

$$\frac{P_1}{T_1}=\frac{P_2}{T_2}$$

Where $P_1$ and $T_1$ are the pressure and the temperature that that "something" would happen and $P_2$ and $T_2$ are the other pressure and temperature when that something should happen. By putting your numbers there you should get the conditions in which you can replicate that "something" if it's not dependent on other things of course.

If the happenstance of that "something" is related to other parameters(which in most cases it is) you must also put those parameters into the relation:

$$\frac{P_1}{T_1}\times a_1\times b_1\times c_1\times \cdots =\frac{P_2}{T_2}\times a_2\times b_2\times c_2\times \cdots $$

Where $a$, $b$, $c$, etc are other parameters that may determine the condition of the occurrence of the "something".

Hope it helps.

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There is no fixed relation between the pressure $P$ and temperature $T$ at which all things happen. It depends very much on the details of what you want to happen, ie how the event is triggered by pressure or temperature, and how these are related in your particular scenario. Contrary to your assumption, "it does matter what."

For example, for ideal gases the relation is
$P=nRT$
where $n$ is the number-density of molecules and $R$ is a constant. If a certain event is triggered when pressure reaches a particular value $P_0$, regardless of temperature, then the temperature could be increased indefinitely while at the same time number density is reduced, keeping the product $nRT$ below the critical value $P_0$. So in these circumstances the event would not happen.

In the case of crystal growth, this is not so much a triggered event as a rate of reaction. The rate of crystal growth increases with number density $n$ (ie particle concentration) but unlike many reactions decreases as temperature $T$ is raised above a certain point.

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