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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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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