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Bulk Modulus $K = \frac{\Delta P}{-\Delta V/V}$.
- The change in pressure is $\Delta P = \frac{mg}{a}$.
- The fractional change in volume for a sphere is $\frac{\Delta V}{V} = 3\frac{dr}{r}$.
- $K = \frac{mg/a}{3dr/r} \implies \frac{dr}{r} = \frac{mg}{3Ka}$.
Using Raoult's Law: $\dfrac{P^0 - P}{P^0} = x_{\text{solute}}$
$\dfrac{185 - 183}{185} = \dfrac{1.2/M}{(1.2/M) + (100/58)}$
$\dfrac{2}{185} = x_{\text{solute}}$
Since $1.2/M \ll 100/58$, approximate: $x_{\text{solute}} \approx \dfrac{1.2/M}{100/58} = \dfrac{1.2 \times 58}{100M}$
$\dfrac{2}{185} = \dfrac{69.6}{100M}$
$M = \dfrac{69.6 \times 185}{200} = \dfrac{12876}{200} \approx \mathbf{128\ \text{g mol}^{-1}}$
If $x > 0$ and $y > 0$, which of the following is equivalent to $\dfrac{x}{y}\sqrt{\dfrac{y}{x^2}}$?
Simplify step by step:
$\dfrac{x}{y} \cdot \sqrt{\dfrac{y}{x^2}} = \dfrac{x}{y} \cdot \dfrac{\sqrt{y}}{\sqrt{x^2}} = \dfrac{x}{y} \cdot \dfrac{\sqrt{y}}{x} = \dfrac{\sqrt{y}}{y} = \dfrac{1}{\sqrt{y}}$
The answer is $\dfrac{1}{\sqrt{y}}$.
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