Q.What is the excess pressure inside a bubble of soap solution of radius $5.00\ \text{mm}$, given that the surface tension of soap solution at the temperature ($20\,^{\circ}\text{C}$) is $2.50 \times 10^{-2}\ \text{N m}^{-1}$? If an air bubble of the same dimension were formed at depth of $40.0\ \text{cm}$ inside a container containing the soap solution (of relative density $1.20$), what would be the pressure inside the bubble? ($1$ atmospheric pressure is $1.01 \times 10^{5}\ \text{Pa}$).

Concept: Surface Tension — Excess Pressure in a Bubble (Laplace's Law) + Hydrostatic Pressure

For a soap bubble, there are two liquid-air interfaces, so the excess pressure is $P_{\text{excess}} = \frac{4T}{r}$.

Step 1: Excess pressure in the soap bubble

Given $T = 2.50 \times 10^{-2}\ \text{N m}^{-1}$, $r = 5.00\ \text{mm} = 5.00 \times 10^{-3}\ \text{m}$.

Step 2: Pressure inside the air bubble at depth

For an air bubble in liquid (single interface), excess pressure is $\frac{2T}{r} = 10.0\ \text{Pa}$.

Hydrostatic pressure at depth $h = 40.0\ \text{cm} = 0.400\ \text{m}$, relative density $1.20$ so $\rho = 1.20 \times 10^{3}\ \text{kg m}^{-3}$.

$P_{\text{hydro}} = \rho g h = (1.20 \times 10^{3})(9.8)(0.400) = 4.70 \times 10^{3}\ \text{Pa}$ …

Concept First: Why Two Interfaces Matter

The key idea here is the number of surfaces a bubble has. A soap bubble in air has two surfaces — an inner surface and an outer surface — each with its own surface tension. That doubles the excess pressure compared to a simple air bubble in liquid (which has only one interface).

The formula for excess pressure (the pressure difference across a curved interface) comes from the Young–Laplace equation:

For a soap bubble, there are two concentric spherical surfaces, so the total excess pressure is:

For an air bubble inside the liquid (like the submerged case), there is only one interface between the air and the liquid, so:

Step-by-step Solution

1. Excess pressure in the soap bubble (in air)

Given:

• Radius $R = 5.00\ \text{mm} = 5.00 \times 10^{-3}\ \text{m}$
• Surface tension $T = 2.50 \times 10^{-2}\ \text{N/m}$

Using the two-interface formula:

2. Pressure inside the submerged air bubble

Now the bubble is formed inside the soap solution at a depth of $40.0\ \text{cm}$. The external pressure on the bubble is not just atmospheric — it also includes the weight of the liquid column above it.

Step 2a: Find the external pressure at that depth

Relative density of the soap solution = $1.20$, so its density is:

Depth $h = 40.0\ \text{cm} = 0.400\ \text{m}$

Hydrostatic pressure:

$= 1200 \times 3.92 = 4704\ \text{Pa}$ …

Part 1: soap bubble, two surfaces, deltaP=4T/r=20.0 Pa. Part 2: submerged air bubble, one surface, deltaP=2T/r=10.0 Pa, plus hydrostatic rhogh~=4.70e3 Pa. Total inside=P_atm+P_hydro+deltaP~=1.06e5 Pa. Key disti …