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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?

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How have you tried to figure it out yourself? What part of the calculation, exactly, are you confused about? – David Z Feb 24 '12 at 20:33
BTW--as it stands the text of the question makes no mention of a phase changing process. Zassounotsukushi has assumed that you are going to freeze the water, but a boiler mechanic might just as well assume you plan to heat it to steam... – dmckee Feb 24 '12 at 21:47

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.

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