The force of a phase transition At standard temperature and pressure, I fill a bottle to capacity with $N$ liters of water, then place a weight of mass $M$ kg on its opening to serve as a lid.  What values of $N$ and $T$, where $T$ is the temperature of the bottle, are sufficient to raise the lid?
 A: Easy!  Any value of $T$ will suffice.  (unless it's ice in a certain temperature range)
Since it's probably a reasonable expectation that you're talking about liquid, subcooled, water for the duration of the problem this is nothing more than multiplication.  The mass of the water is invariant from state $1$ to state $2$ at a higher temperature.
$$M = V \rho(T) $$
Then compute the difference in volume, here $\rho_f$ is the density of saturated fluid.  That is an approximate way to find the density of water by neglecting the compression effect due to pressure.
$$\Delta V = M_2 - M_1 = V \left( \rho(T_1) - \rho(T_2) \right) \approx \left. V \frac{d\rho_f}{dT} \right|_{T_1}$$
Divide by area to find the distance it rises.
$$\Delta z = \left. \frac{V}{A} \frac{d\rho_f}{dT} \right|_{T_1} $$
This change will be positive provided that the derivative is positive.  The derivative is positive for the vast majority of materials and regions.  A notable exception is where the density vs. temperature for ice reverses for a small temperature region.
