Thermodynamics of supercooled water

Now that it's been freezing outside for the last few days, I experimented a bit with supercooling. I've left a bottle of clean water outside for a few hours, and behold, when I shook the bottle, the liquid began to freeze rapidly, as expected.

However, not all of the water froze, only about half of it. This got me thinking.

I reasoned that happens because freezing releases heat, and that heat raises the temperature of the ice and remaning water to 0$^\circ$C, at which point the freezing stops.

To try to quantify this, set

• $H_f = 334000 \frac{J}{kg}$, the "freezing heat" (enthalpy of fusion) of ice,


• $c_i = 2110 \frac{J}{kg\cdot K}$, specific heat of ice,


• $c_w = 4181 \frac{J}{kg\cdot K}$, specific heat of water,


• $M$, initial mass of supercooled water,


• $T_0$, initial temperature of supercooled water,



• $m$, mass of ice at a given moment,


• $T$, temperature at a given moment.

We suppose that the energy released by the freezing of a $\textrm dm$ mass of ice is used up by raising the temperature of both the already frozen ice ($m$) and the remaining water ($M-m$):

$$ \textrm dm \cdot H_f = c_i m \cdot\textrm dT + (M-m)c_w \cdot\textrm dT$$

By rearranging we get a linear differential equation for $m(T)$:

$$\frac{\textrm dm}{\textrm dT}+\frac{c_w-c_i}{H_f}m = \frac{c_w}{H_f}M$$

The solution of this equation, using the initial condition that at $T=T_0$, $m=0$:

$$ m = M \frac{c_w}{c_w-c_i}\left[ 1-e^{-\frac{c_w-c_i}{H_f}(T-T_0)} \right]$$

This looks neat, however, plugging in the actual values of the constants, $T = 0^\circ$C and $T_0 = -6^\circ$C, I get that little more than 7% of the water should freeze - as opposed to more than half, as I've observed. What's worse, the formula can be used to calculate the starting temperature required to freeze all the supercooled water: $$T_{all} = \frac{H_f \log\frac{c_w}{c_i}}{c_i-c_w}=-110.26^\circ \textrm C$$ which seems absurd.

Where is the error in the calculation above?

I've made a number of assumptions which may or may not hold:

• the constants are really constant. I think this is a good approximation for $c_w$, a not that good but still acceptable approximation for $c_i$, and I suppose it holds to $H_f$ to some extent as well.


• there is no heat transfer with the environment, i.e. the bottle is a closed system. The process seemed fast enough (a few seconds at most) for this to be a good approximation.


• the system is in a thermal equilibrium at all times and the heat released by the freezing of the ice is distributed evenly in the emerging ice-water mixture. Now this is where I have doubts. Precisely because of the process happening so fast, this might not be true at all. However, acknowledging the presence of temperature variations makes the problem order of magnitudes harder: one has to solve a system of partial differential equations that take heat transfer, the shape of the bottle and \$deity knows what else into account.