Question

The volume of an air bubble is doubled as it rises from the bottom of a lake to its surface. The atmospheric pressure is 75 cm of mercury and the ratio of the density of mercury to that of lake water is 40/3. The depth of the lake is:

• A. 15 m
• B. 10 m
• C. 20 m
• D. 25 m

Hint:

Boyle’s law helps relate the pressure and volume of a gas at constant temperature. The pressure and volume are inversely proportional.

Answer 1

The Correct Option is C

Step 1: By Boyle's law with the bubble's volume doubling at the surface, the absolute pressure at the bottom is twice the atmospheric pressure: \( P_1 = 2P_2 \), and since \( P_1=P_2+\rho_w gh \), this gives \( P_2=\rho_w gh \).

Step 2: Convert the atmospheric pressure from 75 cm of mercury to pascals: \( P_2 = \rho_{Hg}\,g\,(0.75\,\text{m}) = 13600\times9.8\times0.75 \approx 99960\ \text{Pa} \).

Step 3: Find the density of water from the given ratio: \( \rho_w = \rho_{Hg}\big/(40/3) = 13600\times(3/40) = 1020\ \text{kg/m}^3 \).

Step 4: Solve \( P_2=\rho_w gh \) for \( h \): \( h = \dfrac{99960}{1020\times9.8} \approx 10.0\ \text{m} \).\[ \boxed{h \approx 10\ \text{m}} \]

Answer 2

Instead of converting anything to pascals, keep every pressure expressed as an equivalent height of mercury. Atmospheric pressure is already given as \( 75\,\text{cm Hg} \).

From Boyle's law, doubling the bubble's volume at the surface means the pressure at the bottom is twice atmospheric, so the extra pressure supplied by the water column alone must equal one atmosphere, i.e. also worth \( 75\,\text{cm Hg} \) in mercury-equivalent units.

To convert that 75 cm of mercury into an equivalent height of water, use the fact that a shorter column of the denser fluid balances a taller column of the lighter one, in inverse proportion to density: \( \rho_{Hg}\,(0.75\,\text{m}) = \rho_w\,h \), so:

\[ h = 0.75\,\text{m}\times\frac{\rho_{Hg}}{\rho_w} = 0.75\times\frac{40}{3} \]

1. Option A, 15 m: \( 0.75\times\frac{40}{3}\ne15 \); ruled out.
2. Option B, 10 m: \( 0.75\times\frac{40}{3}=10 \), matching exactly.
3. Option C, 20 m: would double the correct depth and does not follow from this ratio.
4. Option D, 25 m: also does not follow from the 40/3 ratio applied to 0.75 m.

The correct answer is 10 m.