Predict/Calculate Mantles in Gas Lanterns Gas lanterns used...

Predict/Calculate Mantles in Gas Lanterns Gas lanterns used on camping trips have mantles (small lacy bags that give off light) made from a rayon mesh impregnated with thorium and other materials. Thorium is used, even though it is radioactive, because it forms an oxide that can withstand being incandescent for long periods of time. Almost all natural thorium is $_{90}^{232}$ Th, which has a half-life of $1.405 \times 10^{10} \mathrm{y} .$ (a) A typical mantle contains 325 $\mathrm{mg}$ of thorium. What is the activity of the mantle? (b) If the half-life of thorium had been double its actual value, by what factor would the activity of a mantle be changed? Explain.

Predict/Calculate Mantles in Gas Lanterns Gas lanterns used on camping trips have mantles (small lacy bags that give off light)
made from a rayon mesh impregnated with thorium and other
materials. Thorium is used, even though it is radioactive, because it forms an oxide that can withstand being incandescent for long
periods of time. Almost all natural thorium is $_{90}^{232}$ Th, which has a
half-life of $1.405 \times 10^{10} \mathrm{y} .$ (a) A typical mantle contains 325 $\mathrm{mg}$ of thorium. What is the activity of the mantle? (b) If the half-life
of thorium had been double its actual value, by what factor would
the activity of a mantle be changed? Explain.

Key Concepts

Decay Constant

The decay constant is a parameter that represents the probability per unit time that a given nucleus will decay. It is mathematically related to the half-life through the equation ? = ln(2)/t½, linking the exponential decay governing radioactive processes to observable measures such as activity.

Activity

Activity, in the context of nuclear physics, refers to the number of radioactive decays per unit time, typically measured in becquerels (Bq) or curies (Ci). It provides a quantitative measure of the level of radioactivity of a sample and is directly proportional to both the number of radioactive nuclei present and the decay constant.

Relationship Between Activity and Half-life

The activity of a radioactive sample is dependent not only on the number of radioactive atoms present but also on the decay constant. When the half-life changes, this alters the decay constant and hence the activity. Specifically, if the half-life is doubled, the decay constant is halved, leading to a proportional decrease in activity, assuming the number of radioactive atoms remains constant.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation, such as alpha, beta, or gamma rays. This phenomenon occurs spontaneously and can be characterized mathematically as an exponential decay process, with the number of undecayed nuclei decreasing over time according to a fixed probability per unit time.

Half-life

The half-life of a radioactive substance is the time required for half of the initially present nuclei to decay. It is a critical measure that helps in understanding the rate of decay and the persistence of radioactivity, and it can be used to compare the properties of different radioactive materials.

Transcript

00:01 Question 75 deals with gas lanterns.

00:05 Those that are used on camping trips have mantles made of a rayon mesh impregnated with thorium and other materials.

00:12 Thorium is used even though it is radioactive because it forms an oxide that can withstand being incandescent for long periods of time, meaning it'll be lit for a long period of time.

00:24 So while most thorium, natural thorium, 232, 32, sorry, 90, what we're dealing with, has a lot of a half -life of 1 .405 times 10 to 10 years.

00:54 A typical mantle contains 325 milligrams of thorium.

00:57 What is the activity of the mantle? and part b is related to the part a.

01:01 So we'll just go through part a first.

01:04 So if the mass we're given is 325 milligrams, a very small amount, we want to find the activity.

01:16 So the activity that we care about is based on the decay factor times the number of particles that are or nuclei or atoms, whatever you're dealing with.

01:30 So because we're given a mass, we can determine how many atoms are in the sample.

01:39 So to do that, we know that first of all, our decay constant is simply the natural algorithm of two over our half -life.

01:53 And then to solve for the number of atoms, since we're given a mass, we've mass over our molar mass, which we can look up, remember, molar mass is in grams per mole.

02:08 So this equation here, just to check our units, we have grams on top, which is what's given over a molar mass, which is grams per mole.

02:17 We can multiply that by avogadro's number in order to determine the number of atoms per mole.

02:27 So our one of our moles cancel, our grams cancel and we are left with just atoms.

02:32 So this ratio will give us the amount of atoms present.

02:39 We can start plugging in numbers now.

02:43 So we're natural algorithm of 2 over t 1 .5, which is 1 .405 times 10 to 10 years.

02:55 So we want an activity...

Please give Ace some feedback