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Chapter 7: Problem 995

8000 identical water drops are combined to form a bigdrop. Then the ratio of the final surface energy to the initial surface energy of all the drops together is (A) \(1: 10\) (B) \(1: 15\) (C) \(1: 20\) (D) \(1: 25\)

Short Answer

Expert verified
The ratio of the final surface energy to the initial surface energy of all the drops together is 1:20, which corresponds to answer choice (C).

Step by step solution

01

Write the formula for the surface area of a sphere

We will use the formula for the surface area of a sphere to compare the surface energies. The surface area of a sphere with radius r is given by: \[A_s = 4\pi r^2\]
02

Find the initial surface area of the 8000 individual drops

Assuming each drop has a radius of r, the total surface area of the 8000 individual drops would be: \[A_{initial} = 8,000 \times 4\pi r^2\]
03

Find the volume of the big drop

Since the volumes of the drops are conserved when combined, we can find the volume of the big drop by multiplying the volume of a single drop by 8000. The volume of a sphere with radius r is given by: \[V = \frac{4}{3} \pi r^3\] So the volume of the big drop will be: \[V_{big} = 8,000 \times \frac{4}{3} \pi r^3\]
04

Find the radius of the big drop

To find the radius of the big drop, we can set the volume of the big drop equal to the volume of a sphere with radius R: \[8,000 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi R^3\] Solving for R, we get: \[R = \sqrt[3]{8,000 \times r^3}\]
05

Find the final surface area of the big drop

We can now use the radius R of the big drop to find its surface area: \[A_{final} = 4\pi R^2\] Substitute the expression for R from step 4: \[A_{final} = 4\pi (\sqrt[3]{8,000 \times r^3})^2\]
06

Find the ratio of the final surface energy to the initial surface energy

Now we can find the ratio of the surface energies by taking the ratio of the final surface area to the initial surface area: \[\frac{A_{final}}{A_{initial}} = \frac{4\pi (\sqrt[3]{8,000 \times r^3})^2}{8,000 \times 4\pi r^2}\] Simplify the expression, and we get: \[\frac{A_{final}}{A_{initial}} = \frac{(\sqrt[3]{8,000})^2}{8,000}\] Since \(\sqrt[3]{8,000} = 20\), the ratio becomes: \[\frac{A_{final}}{A_{initial}} = \frac{20^2}{8,000} = \frac{1}{20}\] So the ratio of the final surface energy to the initial surface energy of all the drops together is 1:20, which corresponds to answer choice (C).

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