/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 A soap bubble in vacuum has a ra... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 21

A soap bubble in vacuum has a radius of \(3 \mathrm{~cm}\) and another soap bubble in vacuum has a radius of \(4 \mathrm{~cm}\). If two bubbles coalesce under isothermal conditions, then the radius of the new bubble is (A) \(2.3 \mathrm{~cm}\) (B) \(4.5 \mathrm{~cm}\) (C) \(5 \mathrm{~cm}\) (D) \(7 \mathrm{~cm}\)

Short Answer

Expert verified
The radius of the new bubble when the given two soap bubbles coalesce under isothermal conditions is approximately \(4.5 \mathrm{~cm}\).

Step by step solution

01

Understand the pressure difference due to surface tension

Inside a soap bubble, there is a pressure difference between the inside and outside of the bubble due to surface tension. This pressure difference can be expressed as: \[P = 2 * \frac{T}{R}\] Where \(P\) is the pressure difference, \(T\) is the surface tension, and \(R\) is the radius of the bubble.
02

Write the expression for the volume of each bubble

Since the bubbles have radii \(3 \mathrm{~cm}\) and \(4 \mathrm{~cm}\), we can compute their volumes using the formula for the volume of a sphere: \[V_1 = \frac{4}{3} * \pi * (3 \mathrm{~cm})^3\] \[V_2 = \frac{4}{3} * \pi * (4 \mathrm{~cm})^3\]
03

Calculate the total volume of the final bubble

The final bubble is formed by conserving the total volume of the two initial bubbles: \[V_{total} = V_1 + V_2\]
04

Write the expression for the volume of the final bubble

Let \(R_{new}\) be the radius of the new bubble. The volume of the final bubble can be expressed as: \[V_{new} = \frac{4}{3} * \pi * R_{new}^3\] Since \(V_{new} = V_{total}\), we can set these two expressions equal to each other to solve for \(R_{new}\).
05

Solve for R_{new}

Using the volume equations set equal to one another, we can solve for \(R_{new}\): \[\frac{4}{3} * \pi * R_{new}^3 = \frac{4}{3} * \pi * (3 \mathrm{~cm})^3 + \frac{4}{3} * \pi * (4 \mathrm{~cm})^3\] Divide both sides by \(\frac{4}{3} * \pi\): \[R_{new}^3 = (3 \mathrm{~cm})^3 + (4 \mathrm{~cm})^3\] \[R_{new}^3 = 27\mathrm{cm^3} + 64\mathrm{cm^3}\] \[R_{new}^3 = 91\mathrm{cm^3}\] Now take the cube root of both sides: \[R_{new} = \sqrt[3]{91\mathrm{cm^3}}\] R_{new} is approximately \(4.5 \mathrm{~cm}\). Therefore, the correct answer is (B) \(4.5 \mathrm{~cm}\).

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