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The strength of a hydrogen bond in water at ambient conditions is about $8kT$, where $k$ is the Boltzmann constant and $T$ is the temperature. So why do hydrogen bonds form and break so readily in water when their energy is so much higher than $kT$? And what determines the timescale $(\sim\, 1 - 10\ \rm ps)$ of the bond making and breaking?

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Using Transition state theory approximation for the rate of water molecule to cross over the bonding barrier, $E_b = 8KT \approx 21 KJ/mole$:

$$ \tag{1} \frac{1}{\tau} = \omega \exp\left\{-\frac{E_b}{KT}\right\}. $$

Now, we have to estimate the frequency prefactor $\omega$ in Eq.(1). Since there is no data available for the vibration between water-water molecule interaction (don't mess up with the vibration within the water molecule), we simply simulate a quadratic potential from the bond length and bond energy: \begin{align} \frac{1}{2} k x^2 =& E_b;\\ k =& \frac{2 E_b}{x^2};\,\,\,\text{using } E_b= 21 \text{ KJ/mole, and bond length } 3 \dot A;\\ \omega = & \sqrt{\frac{k}{m}}=\sqrt{\frac{2 E_b}{m x^2}} = \sqrt{\frac{2 (21\times 10^3)/(6\times10^{23})}{0.018/(6\times 10^{23}) \times (3\times 10^{-10})^2}} = 5\times 10^{12}\frac{1}{s} \end{align}

Using this estimation into Eq.(1), the rate of crossing the energy barrier: $$ \frac{1}{\tau} = 5\times 10^{12} \exp(-8) = 2\times 10^9. $$ Therefore the bond breaking time scale is about $$ \tau =\frac{1}{2} 10^{-9} sec \approx 0.5 \, ns. $$

In the article ACS, where they estimate the activation energy to be $11$ KJ/mole (approximate $4\,KT$) half of the binding energy I adopted, which brings a factor of $e^4=55$ to the estimation here. This will render a bond breaking time to $9$ ps. Their low activation energy is due to large fluctuation of the hydrogen bonding energy. It arises from the frequent drawn near of two oxygen molecules, which generate a pulse of repulsion force to break the bond, results in a rather low activation energy.

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  • $\begingroup$ Your calculated lifetime seems a bit long, but in any case, what is the explanation to my other question -- why do hydrogen bonds break and form so readily in water given that the bond strength is $8kT$? $\endgroup$
    – user27771
    Commented Apr 14, 2021 at 13:40
  • $\begingroup$ Where did you come up with the fast bond breaking time? $\endgroup$
    – ytlu
    Commented Apr 14, 2021 at 13:41
  • $\begingroup$ Might be even less than 1 ps according to this reference: pubs.acs.org/doi/10.1021/jp407768u $\endgroup$
    – user27771
    Commented Apr 14, 2021 at 13:47
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    $\begingroup$ $k_B T$ at $300 K$ is $ 25 meV$ and in units of $RT$ is $2.5 \frac{KJ}{mol}$ , so $8k_B T$ in units of $RT$ are $ 20\frac{ KJ}{mol} $ $\endgroup$
    – Mark_Bell
    Commented Apr 14, 2021 at 13:50
  • $\begingroup$ Yes. I mess up with the Kcal with KJ, it shoud be 21 KJ.mole. $\endgroup$
    – ytlu
    Commented Apr 14, 2021 at 14:04

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