At what absolute temperatures do molecules occupy each energy level?

1 Answer
Truong-Son N.
Jul 19, 2017

At absolute what temperature? Absolute temperature is in units of #"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 #"0 K"#.
  • All molecules have ONLY vibrational energy at #"0 K"#.
  • As you increase the temperature from #"0 K"#, molecules begin to gain rotational energy (usually before #"100 K"#), and at really high temperatures (a few thousand #"K"#), they gain vibrational energy.
  • At really, really, really high temperatures (tens of thousands of #"K"#, probably), they gain electronic energy.

DISCLAIMER: LONG ANSWER!

The most common different kinds of energies are:

  • translational
  • rotational
  • vibrational
  • electronic

These energy levels are in general differently spaced as follows:

#"electronic"# #">>"# #"vibrational"# #># #"rotational"# #">>"# #"translational"#

http://www.4college.co.uk/

For the temperatures that we care about, these temperatures are called the rotational and vibrational temperatures, #Theta_(rot)# and #Theta_(vib)#, but I suppose we can define an "electronic temperature" #Theta_(el ec)#.

We have:

#Theta_(rot) = (B_e)/(k_B)# in units of #"K"#, usually in the tens of #"K"#

#Theta_(vib) = (omega_e)/(k_B)# in units of #"K"#, usually in the hundreds or thousands of #"K"#

#Theta_(el ec) = (T_e)/(k_B)# in units of #"K"#, usually in the upper thousands or tens of thousands of #"K"#

where:

  • #B_e# is the rotational constant in #"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 #"cm"^(-1)# for each vibration (the number of vibrational motions are #3N - 5# for linear polyatomic molecules and #3N - 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 ~~ "0.695 cm"^(-1)"/K"# is the Boltzmann constant in #"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 #"0 K"#.
  • All molecules have nonzero vibrational energy at #"0 K"#.
  • As you increase the temperature from #"0 K"#, molecules begin to gain rotational energy (usually before #"100 K"#), and at really high temperatures (a few thousand #"K"#), they gain vibrational energy.
  • At really, really, really high temperatures (tens of thousands of #"K"#, probably), they gain electronic energy.

EXAMPLE MOLECULE: CO

As an example (http://webbook.nist.gov/cgi/cbook.cgi?ID=C630080&Units=SI&Mask=1000):

  • #"CO"# has a rotational frequency at around #"1.93 cm"^(-1)#, so only past around

#"1.93 cm"^(-1)/("0.695 cm"^(-1)"/K") ~~ ul"2.78 K"#

would it rotate from natural surrounding heat.

  • #"CO"# has a vibrational frequency at around #"2169.8 cm"^(-1)#, so only past around

#"2169.8 cm"^(-1)/("0.695 cm"^(-1)"/K") ~~ ul"3122 K"#

would it vibrate from natural surrounding heat.

  • #"CO"# has its first excited-state electronic energy level about #"48686.7 cm"^(-1)# above the ground state, which means it won't be until at least

#"48686.7 cm"^(-1)/("0.695 cm"^(-1)"/K") ~~ ul"70000 K"#

or so before #"CO"# has electronic energy naturally from surrounding heat.

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