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Chapter 18: Problem 2472

It the radius of a nucleus of mass number 3 is \(\mathrm{R}\). then the radius of a nucleus of mass number 81 is (A) \(27 \mathrm{R}\) (B) \(9 \mathrm{R}\) (C) \(3 \mathrm{R}\) \((\mathrm{D})(27)^{(1 / 2)} \mathrm{R}\)

Short Answer

Expert verified
The radius of a nucleus with mass number 81 is (C) \(3 \mathrm{R}\).

Step by step solution

01

Write down the formula for the radius of the nucleus with mass number 3.

We know the radius of a nucleus with mass number 3 is R. Using the formula, we have: \( R = R_0 (3)^{\frac{1}{3}} \)
02

Write down the formula for the radius of the nucleus with mass number 81.

We want to find the radius of a nucleus with mass number 81. Using the formula, we have: \( R_n = R_0 (81)^{\frac{1}{3}} \)
03

Divide the equation for mass number 81 by the equation for mass number 3.

By dividing both equations, we can eliminate the constant \(R_0\), allowing us to find the relationship between the radii of the nuclei with mass numbers 3 and 81: \(\frac{R_n}{R} = \frac{R_0 (81)^{\frac{1}{3}}}{R_0 (3)^{\frac{1}{3}}} \)
04

Simplify the equation.

The constant \(R_0\) can now be cancelled out, and we can simplify the expression as follows: \(\frac{R_n}{R} = \frac{(81)^{\frac{1}{3}}}{(3)^{\frac{1}{3}}} = (27)^{\frac{1}{3}} = 3\)
05

Find the desired radius.

Finally, we multiply both sides of the equation by the given radius R: \(R_n = 3R\) Comparing our result with the given options, we find that the correct answer is (C) \(3 \mathrm{R}\).

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Most popular questions from this chapter

In the nuclear reaction \(\mathrm{X}(\eta, \alpha)_{3}^{7}\) Li the atom \(\mathrm{X}\) will be (A) \({ }_{2} \mathrm{He}^{4}\) (B) \(_{5} \mathrm{~B}^{11}\) (C) \(_{5} \mathrm{~B}^{10}\) (D) \({ }_{5} \mathrm{~B}^{9}\)

A nucleus of \({ }^{210}{ }_{84}\) Po originally at rest emits \(\alpha\) -particle with speed \(\mathrm{v}\) what will be the recoil speed of the daughter nucleus (A) \([\mathrm{v} /(214)]\) (B) \([(4 \mathrm{v}) /(214)]\) (C) \([(4 \mathrm{v}) /(206)]\) (D) \([\mathrm{v} /(206)]\)

The Rutherford revolution Per second made by an electron in the first Bohr orbit of hydrogen atom is of the order of (A) \(10^{15}\) (B) \(10^{20}\) (C) \(10^{10}\) (D) \(10^{19}\)

starting with a sample of Pure \({ }^{66} \mathrm{cu},(7 / 8)\) of it decays into \(\mathrm{Zn}\), 15 minutes. The half life the sample is (A) \(5 \mathrm{~min}\) (B) \(7.5 \mathrm{~min}\) (C) \(10 \mathrm{~min}\) (D) \(15 \mathrm{~min}\)

The innermost orbit of the hydrogen atom has a radius \(0.53 \mathrm{~A}\). what is radius of \(2^{\text {nd }}\) orbit is ? (A) \(2.12 \AA\) (B) \(1.06 \AA\) (C) \(21.2 \AA\) (D) \(10.6 \AA\)
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