An Air Bubble in Soap Solution

solved SurfaceTension

Q. An air bubble of radius 5.00 mm $r = 5.00$ mm is formed at depth of 40.0 cm $h = 40.0$ cm inside a container containing soap solution (of relative density 1.20 $ρ = 1.20 \times 10^{3}$ kg/m³). The surface tension of soap solution at the temperature (20 °C) is 2.50 × 10⁻² N/m $S = 2.50 \times 10^{−2}$ N/m. Taking 1 atmospheric pressure as 1.01 × 10⁵ Pa $P_{a} = 1.01 \times 10^{5}$ Pa and g as 9.8 m/s² $g = 9.8$ m/s², what would be the pressure inside the bubble, in pascals?

Inputs

radius

r

5 mm

depth

h

40 cm

tension

S

0.025 N/m

atmosphere

P_{a}

1.01e+05 Pa

gravity

g

9.8 m/s²

Outputs

pressure

P_{i}

How it's solved

Solved with SurfaceTension

Given

$r = 5.00$ mm
$h = 40.0$ cm
$S = 2.50 \times 10^{−2}$ N/m
$P_{a} = 1.01 \times 10^{5}$ Pa
$g = 9.8$ m/s²
$ρ = 1.20 \times 10^{3}$ kg/m³ from “relative density 1.20”

Find $P_{i}$ .

Working

$r = 5.00$ mm $= 0.005$ m
$h = 40.0$ cm $= 0.4$ m
pressure of the fluid above(bubble) $P_{g} = ρ \cdot g \cdot h$
$P_{g} = 1200 \cdot 9.8 \cdot 0.4$
$P_{g} = 4704$ Pa
pressure adds the surface s(bubble) $P = P_{a} + P_{g}$
$P = 101000 + 4704$
$P = 105704$ Pa
excess pressure in a bubble in liquid(bubble) $Δ P = \frac{2 \cdot S}{r}$
$Δ P = \frac{2 \cdot 0.025}{0.005}$
$Δ P = 10$ Pa
pressure inside a bubble(bubble) $P_{i} = P + Δ P$
$P_{i} = 105704 + 10$
$P_{i} = 105700$ Pa
$P_{i} \approx 1.06 \times 10^{5}$ Pato 3 significant figures

Conclusion

The air in the bubble bears the atmosphere and the solution above it, and its one curved surface squeezes it a little more.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :an_air_bubble_in_soap_solution do4  kind :numerical5  describes "An air bubble of radius %{radius} is formed at depth of %{depth} inside a container containing soap " \6            "solution (of relative density 1.20). The surface tension of soap solution at the temperature (20 °C) " \7            "is %{tension}. Taking 1 atmospheric pressure as %{atmosphere} and g as %{gravity}, %{pressure}, in " \8            "pascals?"9  given radius: "5.00 mm"10  given depth: "40.0 cm"11  given tension: "2.50e-2 N/m"12  given atmosphere: "1.01e5 Pa"13  given gravity: "9.8 m/s2"14  asks pressure: "what would be the pressure inside the bubble"15end

solution.rb

1require_relative "../questions/an_air_bubble_in_soap_solution"2require_relative "../science/mechanical_properties_of_fluids/surface_tension"34solve :an_air_bubble_in_soap_solution, using: SurfaceTension do5  objects :bubble67  take radius: bubble.radius8  take depth: bubble.depth9  take tension: S10  take atmosphere: P_a11  take gravity: g1213  deduce_from "relative density 1.20", rho => "1.20e3 kg/m3"1415  pressure = bubble.pressure_inside16  answer pressure:, in: "Pa", figures: 31718  conclusion "The air in the bubble bears the atmosphere and the solution above it, and its one curved " \19             "surface squeezes it a little more."20end

playground.rb

1require_relative "../solutions/an_air_bubble_in_soap_solution"23play_with :an_air_bubble_in_soap_solution do4  title "An Air Bubble in Soap Solution"5end

The laws it drew on

quantity	written	units	within	from
mass	$m$	kg	0 … ∞	Pressure
gravity	$g$	m/s²	—	Pressure
thrust	$F$	N	—	Pressure
area	$A$	m²	—	Pressure
pressure	$P$	Pa	0 … ∞	Pressure
pressure at the surface	$P_{a}$	Pa	—	PressureAtDepth
density	$ρ$	kg/m³	0 … ∞	PressureAtDepth
depth	$h$	m	—	PressureAtDepth
gauge pressure	$P_{g}$	Pa	—	PressureAtDepth
surface tension	$S$	N/m	—	SurfaceTension
radius	$r$	m	—	SurfaceTension
excess pressure	$Δ P$	Pa	—	SurfaceTension
excess pressure in a soap bubble	$Δ P_s$	Pa	—	SurfaceTension
pressure inside	$P_{i}$	Pa	—	SurfaceTension
slider length	$l$	m	—	SurfaceTension
pull of the film	$F_{S}$	N	—	SurfaceTension
pressure of ground	$P_{g}$	Pa	0 … ∞
thrust on ground	$F_{g}$	N	—
area of ground	$A_{g}$	m²	—
pressure of bubble	$P$	Pa	0 … ∞
thrust on bubble	$F$	N	—
area of bubble	$A$	m²	—
gauge pressure on ground	$P_{g,g}$	Pa	—
depth of ground	$h_{g}$	m	—
gauge pressure on bubble	$P_{g}$	Pa	—
depth of bubble	$h$	m	—
excess pressure of ground	$Δ P_{g}$	Pa	—
radius of ground	$r_{g}$	m	—
excess pressure of bubble	$Δ P$	Pa	—
radius of bubble	$r$	m	—
excess pressure in a soap bubble of ground	$Δ {P_s}_{g}$	Pa	—
excess pressure in a soap bubble of bubble	$Δ P_s$	Pa	—
pressure inside of ground	$P_{i,g}$	Pa	—
pressure inside of bubble	$P_{i}$	Pa	—
pull of the film on ground	$F_{S,g}$	N	—
slider length of ground	$l_{g}$	m	—
pull of the film on bubble	$F_{S}$	N	—
slider length of bubble	$l$	m	—
mass of bubble	$m$	kg	0 … ∞

P_{s} = \frac{F_{s}}{A_{s}}

pressure is force over area

P_{g,p} = ρ \cdot g \cdot h_{p}

pressure of the fluid above

P_{p} = P_{a} + P_{g,p}

pressure adds the surface s

Δ P_{b} = \frac{2 \cdot S}{r_{b}}

excess pressure in a bubble in liquid

Δ {P_s}_{b} = \frac{4 \cdot S}{r_{b}}

excess pressure in a soap bubble with two surfaces

P_{i,b} = P_{b} + Δ P_{b}

pressure inside a bubble

F_{S,s} = 2 \cdot S \cdot l_{s}

a film pulls with both its faces

P_{g} = \frac{F_{g}}{A_{g}}

pressure is force over area(ground)

P = \frac{F}{A}

pressure is force over area(bubble)

P_{g,g} = ρ \cdot g \cdot h_{g}

pressure of the fluid above(ground)

P_{g} = ρ \cdot g \cdot h

pressure of the fluid above(bubble)

P_{g} = P_{a} + P_{g,g}

pressure adds the surface s(ground)

P = P_{a} + P_{g}

pressure adds the surface s(bubble)

Δ P_{g} = \frac{2 \cdot S}{r_{g}}

excess pressure in a bubble in liquid(ground)

Δ P = \frac{2 \cdot S}{r}

excess pressure in a bubble in liquid(bubble)

Δ {P_s}_{g} = \frac{4 \cdot S}{r_{g}}

excess pressure in a soap bubble with two surfaces(ground)

Δ P_s = \frac{4 \cdot S}{r}

excess pressure in a soap bubble with two surfaces(bubble)

P_{i,g} = P_{g} + Δ P_{g}

pressure inside a bubble(ground)

P_{i} = P + Δ P

pressure inside a bubble(bubble)

F_{S,g} = 2 \cdot S \cdot l_{g}

a film pulls with both its faces(ground)

F_{S} = 2 \cdot S \cdot l

a film pulls with both its faces(bubble)