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Chapter 7: Problem 8

The excess pressure inside a bubble (discussed in Chapter 1 ) is known to be dependent on bubble radius and surface tension. After finding the pi terms, determine the variation in excess pressure if we (a) double the radius and (b) double the surface tension.

Short Answer

Expert verified
If the radius of the bubble is doubled, the excess pressure becomes half of the original excess pressure. If the surface tension is doubled, the excess pressure doubles the initial excess pressure.

Step by step solution

01

Understand the pressure formula

It is known that the excess pressure \(\Delta P\) inside a bubble is given by the formula \(\Delta P = \frac{4 \cdot T}{r}\), where \(T\) is the surface tension and \(r\) is the radius of the bubble.
02

Variation in pressure with doubled radius

If the radius of the bubble is doubled to \(2r\), the pressure should be determined using the given formula. Thus: \(\Delta P' = \frac{4 \cdot T}{2r} = \frac{4 \cdot T}{r} \cdot \frac{1}{2} = \frac{\Delta P}{2}\). So, if the radius is doubled, the excess pressure becomes half.
03

Variation in pressure with doubled surface tension

If the surface tension is doubled to \(2T\), we can also determine the new pressure using the formula: \(\Delta P' = \frac{4 \cdot 2T}{r} = \frac{4 \cdot T}{r} \cdot 2 = 2\Delta P\). Thus, if the surface tension is doubled, the excess pressure doubles.

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