Question

An air bubble of radius 1.0 mm is observed at a depth of 20 cm below the free surface of a liquid having surface tension \( 0.095 \, \text{J/m}^2 \) and density \( 10^3 \, \text{kg/m}^3 \). The difference between pressure inside the bubble and atmospheric pressure is: Given \( g = 10 \, \text{m/s}^2 \).

Hint:

The pressure difference inside a bubble is due to both the surface tension and the liquid pressure at the given depth. Use the given formulas to compute both contributions.

Answer 1

Correct Answer: 1

To determine the pressure differential within the bubble, it is essential to first identify the contributing pressures. An air bubble submerged in a liquid is subjected to hydrostatic pressure, resulting from the liquid column above, and pressure due to surface tension.

1. Hydrostatic Pressure Calculation:
Hydrostatic pressure \(P_{\text{hydro}}\) at a depth \(h\) is calculated as:

\(P_{\text{hydro}} = \rho \cdot g \cdot h\)

where:
\(\rho = 1000 \, \text{kg/m}^3\) (liquid density),
\(g = 10 \, \text{m/s}^2\) (gravitational acceleration),
\(h = 0.2 \, \text{m}\) (depth).

Therefore:

\(P_{\text{hydro}} = 1000 \cdot 10 \cdot 0.2 = 2000 \, \text{Pa}\).

2. Surface Tension Pressure Calculation:
The pressure within a bubble attributed to surface tension, \(P_{\text{surface}}\), is given by:

\(P_{\text{surface}} = \frac{2 \cdot T}{r}\)

where:
\(T = 0.095 \, \text{J/m}^2\) (surface tension),
\(r = 0.001 \, \text{m}\) (bubble radius).

Thus:

\(P_{\text{surface}} = \frac{2 \cdot 0.095}{0.001} = 190 \, \text{Pa}\).

3. Total Internal Bubble Pressure Calculation:
The total pressure within the bubble, \(P_{\text{inside}}\), is:

\(P_{\text{inside}} = P_{\text{atmospheric}} + P_{\text{hydro}} + P_{\text{surface}}\)

To ascertain the pressure difference relative to atmospheric pressure, \(P_{\text{atmospheric}}\) is omitted from the calculation of the difference:

The resultant pressure difference is:

\(P_{\text{diff}} = P_{\text{hydro}} + P_{\text{surface}} = 2000 + 190 = 2190 \, \text{Pa}\).

The discrepancy between the internal bubble pressure and atmospheric pressure is determined to be 2190 Pa. This precise calculation accurately reflects the provided data and confirms the solution.