# At what absolute temperatures do molecules occupy each energy level?

Jul 19, 2017

At absolute what temperature? Absolute temperature is in units of $\text{K}$, kelvins. It depends on what temperature you are at that certain molecules have access to certain kinds of energy levels, and thus can have that kind of energy.

• All molecules have translational energy at most temperatures, except $\text{0 K}$.
• All molecules have ONLY vibrational energy at $\text{0 K}$.
• As you increase the temperature from $\text{0 K}$, molecules begin to gain rotational energy (usually before $\text{100 K}$), and at really high temperatures (a few thousand $\text{K}$), they gain vibrational energy.
• At really, really, really high temperatures (tens of thousands of $\text{K}$, probably), they gain electronic energy.

The most common different kinds of energies are:

• translational
• rotational
• vibrational
• electronic

These energy levels are in general differently spaced as follows:

$\text{electronic}$ $\text{>>}$ $\text{vibrational}$ $>$ $\text{rotational}$ $\text{>>}$ $\text{translational}$

For the temperatures that we care about, these temperatures are called the rotational and vibrational temperatures, ${\Theta}_{r o t}$ and ${\Theta}_{v i b}$, but I suppose we can define an "electronic temperature" ${\Theta}_{e l e c}$.

We have:

${\Theta}_{r o t} = \frac{{B}_{e}}{{k}_{B}}$ in units of $\text{K}$, usually in the tens of $\text{K}$

${\Theta}_{v i b} = \frac{{\omega}_{e}}{{k}_{B}}$ in units of $\text{K}$, usually in the hundreds or thousands of $\text{K}$

${\Theta}_{e l e c} = \frac{{T}_{e}}{{k}_{B}}$ in units of $\text{K}$, usually in the upper thousands or tens of thousands of $\text{K}$

where:

• ${B}_{e}$ is the rotational constant in ${\text{cm}}^{- 1}$ for each type of rotational motion (up to $2$ for linear molecules and $3$ for nonlinear molecules).
• ${\omega}_{e}$ is the fundamental vibrational frequency in ${\text{cm}}^{- 1}$ for each vibration (the number of vibrational motions are $3 N - 5$ for linear polyatomic molecules and $3 N - 6$ for nonlinear polyatomic molecules, where $N$ is the number of atoms).
• ${T}_{e}$ is the electronic frequency constant (or whatever it's called), and is analogous.
• ${k}_{B} \approx \text{0.695 cm"^(-1)"/K}$ is the Boltzmann constant in $\text{cm"^(-1)"/K}$.

And when these temperatures are surpassed, the molecule has that kind of energy.

In summary...

• All molecules have translational energy at most temperatures, except $\text{0 K}$.
• All molecules have nonzero vibrational energy at $\text{0 K}$.
• As you increase the temperature from $\text{0 K}$, molecules begin to gain rotational energy (usually before $\text{100 K}$), and at really high temperatures (a few thousand $\text{K}$), they gain vibrational energy.
• At really, really, really high temperatures (tens of thousands of $\text{K}$, probably), they gain electronic energy.

EXAMPLE MOLECULE: CO

• $\text{CO}$ has a rotational frequency at around ${\text{1.93 cm}}^{- 1}$, so only past around

$\text{1.93 cm"^(-1)/("0.695 cm"^(-1)"/K") ~~ ul"2.78 K}$

would it rotate from natural surrounding heat.

• $\text{CO}$ has a vibrational frequency at around ${\text{2169.8 cm}}^{- 1}$, so only past around

$\text{2169.8 cm"^(-1)/("0.695 cm"^(-1)"/K") ~~ ul"3122 K}$

would it vibrate from natural surrounding heat.

• $\text{CO}$ has its first excited-state electronic energy level about ${\text{48686.7 cm}}^{- 1}$ above the ground state, which means it won't be until at least

$\text{48686.7 cm"^(-1)/("0.695 cm"^(-1)"/K") ~~ ul"70000 K}$

or so before $\text{CO}$ has electronic energy naturally from surrounding heat.