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Chapter 29: Problem 78

In naturally occurring potassium, \(0.0117 \%\) of the nuclei are radioactive \(^{40} \mathrm{K} .\) (a) What mass of \(^{40} \mathrm{K}\) is found in a broccoli stalk containing \(300 \mathrm{mg}\) of potassium? (b) What is the activity of this broccoli stalk due to \(^{40} \mathrm{K} ?\)

Short Answer

Expert verified
Answer: The activity of the broccoli stalk due to potassium-40 is approximately 9.1912 disintegrations per second (dps).

Step by step solution

01

Calculate the mass of potassium-40 in the broccoli stalk

First, let's find out how many grams of radioactive potassium-40 are present in the broccoli stalk. Since 0.0117% of potassium is radioactive potassium-40, we can find the mass of potassium-40 by multiplying the given percentage by the total mass of potassium in the stalk. mass of potassium-40 = \(\frac{0.0117}{100} \times 300\:mg\)
02

Evaluate the mass

Now, let's evaluate the mass of potassium-40 in the broccoli stalk: mass of potassium-40 = \(\frac{0.0117}{100} \times 300\:mg = 0.0351\:mg\) So, the mass of potassium-40 found in the broccoli stalk is 0.0351 mg.
03

Find the activity of potassium-40

To find the activity of the potassium-40 in the broccoli stalk, we will need its specific activity. The specific activity of potassium-40 is 0.262 disintegrations per second (dps) per microgram. First, let's convert the mass of potassium-40 from mg to µg: mass of potassium-40 = \(0.0351\:mg \times \frac{1000\:µg}{1\:mg} = 35.1\:µg\) Now, we can calculate the activity of potassium-40: activity of potassium-40 = specific activity × mass of potassium-40 activity of potassium-40 = \(0.262\dfrac{dps}{µg} \times 35.1\:µg\)
04

Evaluate the activity

Let's evaluate the activity of potassium-40 in the broccoli stalk: activity of potassium-40 = \(0.262 \frac{dps}{µg} \times 35.1\:µg = 9.1912\:dps\) The activity of the broccoli stalk due to potassium-40 is approximately 9.1912 disintegrations per second (dps).

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