# A soap bubble is formed using a mixture of detergent and water. The surface tension of the mixture is #0.030 ##Nm^-1#. If the bubble has a radius of #2# #cm# and atmospheric pressure is #101.2# #kPa#, what is the gauge pressure inside the bubble?

##### 1 Answer

#### Explanation:

Your tool of choice for this problem will be the **Laplace - Young equation**, which allows you to calculate the *Laplace pressure*, i.e. the *pressure difference* that exists between the inside and the outside of a curved surface that acts as a boundary between a liquid region and a gas region

#color(blue)(|bar(ul(color(white)(a/a)Deltap = gamma (1/R_1 + 1/R_2)color(white)(a/a)|)))" "# , where

*surface tension* of the liquid

The idea here is that a *difference in pressure* between the two sides of a *flat* liquid surface gives rise to a force that can only be balanced by the surface tension of the liquid if the surface of the liquid is **curved**.

The great thing about a *spherical shape* such as a bubble is that the two principal radii of curvature are **equal**. This implies tha the Laplace pressure will be equal to

#Deltap = gamma * (1/R + 1/R)#

which is equivalent to

#color(blue)(|bar(ul(color(white)(a/a)Deltap = gamma * 2/Rcolor(white)(a/a)|))) -># the Laplace pressure forspherical shapes

Before plugging in your values to get the Laplace pressure for your soap bubble, make sure that you convert the radius of the bubble from *centimeters* to *meters* by using the conversion factor

#color(purple)(|bar(ul(color(white)(a/a)color(black)("1 m" = 10^2"cm")color(white)(a/a)|)))#

You will thus have

#Deltap = "0.030 N m"^(-1) * 2/(2 * 10^(-2)"m") = "3 N m"^(-2)#

As you know, *Pascal*,

#Deltap = "3 Pa"#

Now, **gauge pressure** essentially tells you the **difference** that exists between the absolute pressure and *ambient air pressure*. In other words, gauge pressure is zero-referenced against normal pressure, i.e. air pressure at sea level, given to you as

In this context, the Laplace pressure will actually be equivalent to the **gauge pressure**. In other words, the pressure **inside** the soap bubble will be equal to

#P_"inside bubble" = "101,200 Pa" + "3 Pa" = "101,203 Pa"#

The gauge pressure uses **zero reference**, so you will get a gauge reading of

#P_"gauge" = Deltap = "3 Pa"#