Chapter 10: Problem 46

The \(\beta^{-}\) particles emitted in the decay of \(^{3} \mathrm{H}\) (tritium) interact with matter to create light in a glowin-the-dark exit sign. At the time of manufacture, such a sign contains \(15.0 \mathrm{Ci}\) of \(^{3} \mathrm{H}\). (a) What is the mass of the tritium? (b) What is its activity 5.00 y after manufacture?

Short Answer

Expert verified

The initial mass of tritium in the sign is 4.92 x 10⁻¹¹ g, and the activity of tritium in the sign 5 years after manufacture is 3.75 x 10¹¹ disintegrations per second.

Step by step solution

Gather given information

In this problem, we are given: 1. Initial activity, A₀ = 15.0 Ci (Curie) 2. Half-life of tritium, t₁/₂ = 12.33 years 3. Time elapsed, t = 5.00 years We will also need to use: 1. Avogadro's number, Nₐ = 6.022 x 10²³ atoms/mol 2. Decay constant, λ (which we will calculate) 3. Activity formula: A = A₀e^(-λt) 4. The molar mass of tritium: 3.00 g/mol

Convert activity from Curie to atoms per second

We know that 1 Ci corresponds to 3.7 x 10¹⁰ disintegrations per second (dps). So, we can convert the initial activity (15.0 Ci) to disintegrations per second. Initial activity (A₀) = 15.0 Ci x (3.7 x 10¹⁰ dps/Ci) = 5.55 x 10¹¹ dps

Calculate decay constant λ

The decay constant λ is related to half-life by: λ = ln(2) / t₁/₂ Plug in values: λ = ln(2) / 12.33 = 0.05627 yr⁻¹

Find the initial number of tritium atoms

Using the initial activity and decay constant, we can find the initial number of tritium atoms (N₀) in the sign as follows: A₀ = λN₀ ⇒ N₀ = A₀ / λ N₀ = (5.55 x 10¹¹ dps) / (0.05627 yr⁻¹) = 9.86 x 10¹² atoms

Calculate the initial mass of tritium

To find the mass of tritium, we will first determine the initial number of moles, n: n = N₀ / Nₐ n = (9.86 x 10¹² atoms) / (6.022 x 10²³ atoms/mol) = 1.64 x 10⁻¹¹ mol From the molar mass of tritium (3.00 g/mol), we can find the mass: m = n x Molar mass m = (1.64 x 10⁻¹¹ mol) x (3.00 g/mol) = 4.92 x 10⁻¹¹ g So, the initial mass of tritium (m) in the sign is 4.92 x 10⁻¹¹ g.

Compute the activity after 5 years

To find the activity of tritium after 5 years, we can use the activity formula: A = A₀e^(-λt) Plug in the values: A = (5.55 x 10¹¹ dps)e^(-0.05627 x 5) = 3.75 x 10¹¹ dps After 5 years, the activity (A) of tritium in the sign is 3.75 x 10¹¹ disintegrations per second. In conclusion: a) The mass of tritium in the sign at the time of manufacture is 4.92 x 10⁻¹¹ g. b) The activity of tritium in the sign 5 years after manufacture is 3.75 x 10¹¹ disintegrations per second.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Gather given information

Convert activity from Curie to atoms per second

Calculate decay constant λ

Find the initial number of tritium atoms

Calculate the initial mass of tritium

Compute the activity after 5 years

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Magnetism

Waves Physics

Electromagnetism

Medical Physics

Space Physics

Study anywhere. Anytime. Across all devices.