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Chapter 18: Problem 64

A sample of francium- 212 will decay to one sixteenth its original amount after \(80 \mathrm{~min}\). What is the half-life of francium-212?

Short Answer

Expert verified
The half-life of francium-212 is 4 minutes.

Step by step solution

01

Understand the problem

Given that a sample of francium-212 will decay to one-sixteenth its original amount after 80 minutes. Our aim is to find the half-life of this radioisotope. Francium-212 follows exponential decay, so we can use the formula \[ N(t) = N(0) \times (1/2)^{t/T} \] where \( N(t) \) is the remaining amount after time \( t \), \( N(0) \) is the original amount, \( T \) is the half-life we're looking for, and \( t \) is the lapse time, which is 80 minutes.
02

Set up equation

Since our remaining amount is one-sixteenth of the original after 80 minutes, we can plug in the values into our equation. So, our equation becomes \[ 1/16 = (1/2)^{80/T} \].
03

Simplify the equation

Using properties of exponents, we rewrite the left-hand side of the equation, \( 1/16 = 1/2^4 = (1/2)^{4 \cdot 1} = (1/2)^{80/(T/4)} \]. So now we have our equation as \[ (1/2)^{80/(T/4)} = (1/2)^{80/T} \]. When the bases are equal, the exponents must also be equal, thus we get \( 80/(T/4) = 80/T \).
04

Solve for T

Cross multiply to solve for \( T \), we get \( 80T = 4 \cdot 80 \) or \( T = 4 \cdot 80 / 80 = 4 \) minutes. This implies that the half-life of francium-212 is 4 minutes.

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Most popular questions from this chapter

After 4797 y, how much of an original 0.450 g of radium-226 remains? The half-life of radium-226 is 1599 y.

Explain the difference between fission and fusion.

How many milligrams remain of a 15.0 \(\mathrm{mg}\) sample of radium-226 after 6396 \(\mathrm{y} ?\) The half-life of this isotope is 1599 \(\mathrm{y}\) .

Calculating the Amount of Radioactive Material The graphing calculator can run a program that graphs the relationship between the amount of radioactive material and elapsed time. Given the half-life of the radioactive material and the initial amount of material in grams, you will graph the relationship between the amount of radioactive material and the elapsed time. Then, with the elapsed time, you will trace the graph to calculate the amount of radioactive material. Go to Appendix C. If you are using a TI-83 Plus, you can download the program RADIOACT and run the application as directed. If you are using another calculator, your teacher will provide you with key-strokes and data sets to use. After you have run the program, answer these questions. a. Determine the amount of neptunium-235 left after 2.0 years, given the half- life of neptunium-235 is 1.08 years and the initial amount was 8.00 g. b. Determine the amount of neptunium-235 left after 5.0 years, given the half- life of neptunium-235 is 1.08 years and the initial amount was 8.00 g. c. Determine the amount of uranium-232 left after 100 years, given the half- life of uranium-232 is 69 years and the initial amount was 10.0 g.

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