# Hydrogen bonding as a function of $kT$

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?

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.

• 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 $8kT$? Apr 14, 2021 at 13:40
• Where did you come up with the fast bond breaking time?
– ytlu
Apr 14, 2021 at 13:41
• Might be even less than 1 ps according to this reference: pubs.acs.org/doi/10.1021/jp407768u Apr 14, 2021 at 13:47
• $k_B T$ 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 $20\frac{ KJ}{mol}$ Apr 14, 2021 at 13:50
• Yes. I mess up with the Kcal with KJ, it shoud be 21 KJ.mole.
– ytlu
Apr 14, 2021 at 14:04