# 5.9: Most Molecules are in the Ground Vibrational State

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The vibrational energy levels of a diatomic are given by

\[E_v = (v +1/2) h \nu\]

where is \(\nu\) the vibrational frequency and \(v\) is the vibrational quantum number. In this case, it is easy to sum the geometric series shown below

\[ q_{vib} = \sum_{v=0}^{\infty} e^{-( v + 1/2) h\nu / k T} \]

\[ = e^{-h \nu/ 2k_BT} \left( 1 + e^{-h \nu/ 2k_BT} + e^{-2 h \nu/ 2k_BT} + ... \right) \]

or rewritten as

\[ q_{vib} = e^{-h \nu / 2k_BT} \left( 1 +x + x^2 +x^3 + ... \right) \label{eq0}\]

where \(x = e^{-h \nu/k_BT} \).

Given the following power series expansion

\[ \dfrac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + .... \]

Equation \(\ref{eq0}\) can be rewritten as

\[ q_{vib} = e^{-h \nu / 2k_BT} \left( \dfrac{1}{1-x} \right) \]

or

\[q_{vib} = \dfrac{e^{-h \nu / 2 k_B T}} {1 - e^{-h \nu} / k_B T} \label{eq1} \]

If the zero of energy scale is at \(h \nu /2k_BT\), then Equation \(\ref{eq1}\) can be rewritten as

\[q_{vib} \approx \dfrac{1} {1 - e^{-h \nu} / k_B T} \label{VIBQ}\]

A vibrational temperature \(Θ_{vib}\) may be defined as

\[ Θ_{vib}= \dfrac{ hc \tilde{\nu}}{k}\]

where \(\tilde{\nu}\) is the vibrational frequency in cm^{-1}.

\(Θ_{vib}\) is a good way to express the *stiffness* of the vibrating bond in units of the Boltzmann constant. Because the stiffness obviously depends on what bond your are talking about, this is a good way to do the same thing we did for the critical temperature of the non-ideal gases.

Molecule | g | Bond Length (Angstrom) | \(ω\) (cm^{-1}) |
\(Θ_{vib}\) (K) | \(\tilde{B}\) (cm^{-1}) |
\(Θ_{rot}\) (K) | Force constant \(k\) (dynes/cm) | \(D_0\) (kcal/ mol) |
---|---|---|---|---|---|---|---|---|

\(H_2\) | 1 | 0.7474 | 4400 | 6332 | 60.9 | 87.6 | 5.749 | 103.2 |

\(D_2\) | 1 | 0.7415 | 3118 | 4487 | 30.45 | 43.8 | 5.77 | 104.6 |

\(N_2\) | 1 | 1.097 | 2358 | 3393 | 2.001 | 2.99 | 22.94 | 225.1 |

\(O_2\) | 3 | 1.207 | 1580 | 2274 | 1.446 | 2.08 | 11.76 | 118.0 |

\(Cl_2\) | 1 | 1.987 | 560 | 805 | 0.244 | 0.351 | 3.2 | 57.0 |

\(CO\) | 1 | 1.128 | 2170 | 3122 | 1.931 | 2.78 | 19.03 | 255.8 |

\NO\) | 2 | 1.15 | 190 | 2719 | 1.695 | 2.45 | 15.7 | 150.0 |

\(HCl\) | 1 | 1.275 | 2938 | 4227 | 10.44 | 15.02 | 4.9 | 102.2 |

\(HI\) | 1 | 1.609 | 2270 | 3266 | 6.46 | 9.06 | 3.0 | 70.5 |

\(Na_2\) | 1 | 3.096 | 159 | 229 | 0.154 | 0.221 | 0.17 | 17.3 |

\(K_2\) | 1 | 3.979 | 92.3 | 133 | 0.0561 | 0.081 | 0.10 | 11.8 |

## Vibrational Heat Capacity

The vibrational energy is given by the above expression and the molar heat capacity at constant volume, \(\bar{C}_V\) is given by

\[ C_V= \left(\dfrac{∂E}{∂T} \right)_V\]

We have,

\[ \dfrac{∂}{∂T} = \dfrac{∂β}{∂T} \dfrac{∂}{∂β} = \dfrac{-1}{kT^2} \dfrac{∂}{∂β} = (-k β^2 ) (\dfrac{∂}{∂β}) \label{3.54} \]

Therefore,

\[\bar{C}_V = (-k β^2 ) \left(\dfrac{∂ ε_{vib}}{∂β}\right)\]

and when the vibrational partition function (Equation \(\ref{VIBQ}\)) is introduced

\[= -k \beta^2 \dfrac{ \left[ (1 - e^{-hc \nu / k_B T}) (-hc \tilde{\nu}) - e^{-hc \nu / k_B T} (+ hc\tilde{\nu}) \right] e^{-hc \nu / k_B T} }{ (1 - e^{-hc \nu / k_B T})^2 } hv \tilde{\nu}\]

\[= k_B \left( \dfrac{ Θ_{vib} }{T} \right)^2 \dfrac{ e^{- Θ_{vib} /T} }{ \left( 1- e^{- Θ_{vib}/T} \right)^2 } \label{FinalQ}\]

For large \(T\), the \(\bar{C}_V\) becomes

\[N_A k = R\]

and for small T, \(\bar{C}_V\) goes to zero as demonstrated in Figure \(\PageIndex{1}\).

The vibrational heat capacity is shown as function of the reduced temperature T/Θ to get a general picture valid for all diatomic gases. Compare the parameters on Table \(\PageIndex{1}\) to see on what different absolute scales we have to think for different gases. Clearly the vibrational contribution to the heat capacity *depends on temperature*. For many molecules (especially light ones), the vibrational contribution only kicks in at quite high temperatures.

The value of \(Θ_{vib}\)

is determined mostly by- the strength of a bond (the stronger the higher \(Θ_{vib}\) )
- the (effective) mass of the molecule (the lighter the higher \(Θ_{vib}\) )

Molecules with low \(Θ_{vib}\) often dissociate at higher temperatures, although the *harmonic* oscillator model is not sufficient to describe that phenomenon.