Back To Course

MHT-CET: Practice & Study Guide49 chapters | 398 lessons | 39 flashcard sets

Are you a student or a teacher?

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Stephanie Bryan*

Stephanie has a master's degree in Physical Chemistry and teaches college level chemistry and physics.

In this lesson, we'll talk about first-order reactions like radioactive decay. We'll use the mathematical descriptions of these reactions to discuss their behavior.

How can scientists use uranium deposits to date the age of the earth? No one was alive back then - there weren't even bacteria! Well, it turns out the radioactive decay of Uranium-238 is very predictable and can be described mathematically, using first-order reactions.

The **rate of reaction**, or reaction rate, is the speed at which a reaction progresses. We define this mathematically by measuring the rate at which reactants disappear or products appear, where rate is defined as a derivative with respect to time. In this lesson, we're concentrating on first-order reactions. Because **first-order reaction rates** only depend on the concentration of one reactant, we can define the rate of these reactions as the rate of disappearance of this reactant.

**Rate laws** are equations that mathematically describe this rate. Knowing which variables determine the rate of reaction allows us to draw useful conclusions about that reaction. In this lesson, we'll look at first-order reactions, which depend only on the concentration of one reactant. We'll then use this rate law to derive an equation for the half-life of the reaction.

In this expression, the brackets denote concentration, so A in brackets is the concentration of reactant A. The negative sign indicates that the concentration is going down over time, and *t* is time.

First-order reactions are only dependent on the concentration of one reactant raised to the power of one. In other words, in first-order reactions, the rate is proportional to the concentration of reactant A. If it were proportional to the concentration of two different reactants or to the concentration of reactant A-squared, it would be a second-order reaction. We will leave that discussion for another lesson.

We can turn a proportion into an equation by multiplying by a constant. This constant, *k*, is the **reaction rate coefficient**.

Then, plugging the definition of rate from the previous section into our rate law gives us the **differential rate law**.

Starting with the differential rate law,

we want to move all the concentrations to one side of the equation and move the constant and time to the other side of the equation. It can now be read as:

Then we can integrate each side of the equation:

Once we do that, it will give us the integral representation of the rate law:

With a bit of algebra, this can be represented in another way as well:

The **half-life** of a reaction is the time that it takes to reduce the concentration of a reactant by half. In other words, at this point, the concentration of the reactant is half its initial concentration:

If we plug this into our integrated rate law we would get the following equation for the half-life:

We can see from this equation that the half-life is only dependent on the reaction rate coefficient, *k*, which is a constant. This means that the amount of time it takes to reduce the reactant by half is not concentration-dependent. If we start out with 100% of reactant A, it takes the same time to go from 100% to 50% as it does to go from 50% to 25% and from 25% to 12.5%. We can plot this behavior for carbon-14, which has a half-life of 5,730 years:

Notice that the amount of carbon-14 never quite reaches zero - every 5,730 years, it halves again! Uranium-238 has a half-life of nearly 4.5 billion years! This is why it can be used to date the earth. Looking at how much Uranium deposits on Earth have decayed gives us a good idea about the age of the earth.

Let's look at how this information might be used. If a tree sample has 1.0x10-6 g of carbon -14 when it dies (i.e. when it stops exchanging carbon-14 with the earth), then how much carbon-14 will be left in the sample after 9,000 years?

When solving a problem like this, we begin by looking at what is given or known:

- The sample is carbon-14, which has a half-life of 5,730 years
- The time, t, in which carbon-14 decays, is 9,000 years
- The initial concentration is 1.0x10-6 g per sample

If the half-life of carbon-14 is 5,730 years, then we can determine *k* for carbon-14 from our half-life equation we just looked at:

We can then insert the known quantities into the integral representation of the rate law. When solving problems, it's best to choose the version of the equation which is easiest to solve. In this example, we want to use the equation without a natural log. Note that we can cancel out the units of time in the exponent:

Thus, after 9,000 years, there would be 3.4x10-7 g of carbon-14 left in the sample!

Let's briefly go over what we've learned one at a time.

**Rates of reaction**, or reaction rates, are the speeds at which a reaction progresses. **First-order reaction rates** are only dependent on the concentration of one reactant.

This relationship can be expressed in:

A **differential rate law**:

An **integrated rate law**:

This rate law also results in a **half-life**, or the time that it takes to reduce the concentration of a reactant by half, that isn't concentration-dependent:

Half-lives can be a powerful predictive tool for first order reactions. They're most often used in radioactive decay applications which can range from short-term medical use to long-term geological application.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
6 in chapter 19 of the course:

Back To Course

MHT-CET: Practice & Study Guide49 chapters | 398 lessons | 39 flashcard sets

- Go to Wave Motion

- What is Oxidation? - Definition, Process & Examples 5:25
- What Are Antioxidants? - Definition, Foods & Benefits 3:45
- Reducing Agent: Definition & Examples 5:48
- Sulfuric Acid: Formula, Structure & Properties
- Potassium Chromate: Uses & Formula 5:12
- First-Order Reactions: Definition & Mathematical Representation 7:23
- Reaction Mechanisms and The Rate Determining Step 4:34
- Go to Chemical Reactions & Kinetics

- GRE Information Guide
- Computer Science 310: Current Trends in Computer Science & IT
- Earth Science 105: Introduction to Oceanography
- Computer Science 331: Cybersecurity Risk Analysis Management
- Computer Science 336: Network Forensics
- Practical Applications for Excel
- Practical Applications in Business Law
- Practical Applications for How to Network
- Practical Application for Technical Writing
- Practical Applications for Workplace Communications with Computers
- MEGA Test Accomodations
- Study.com Grant for Teachers
- What are the MEGA Tests?
- MOGEA Test Score Information
- ASWB Prep Product Comparison
- What is the MOGEA Test?
- TASC Exam Registration Information

- Promoting Motor Learning With Activities, Sports & Games
- How to Teach Conflict Resolution to Kids
- Witchcraft in the Jacobean Era
- Best Practices for Teaching Gifted Students
- Practical Application: Principles of Composition in Graphic Design Infographic
- Recursion & Recursive Algorithms in Python: Definition & Examples
- Practical Application: Five Major Consumer Rights Infographic
- Food & Beverage Service & Operations: Assignment 2 - Research Paper
- Quiz & Worksheet - Aristotelian Virtue Ethics
- Quiz & Worksheet - Assessing Diverse Learners
- Quiz & Worksheet - Citizen Journalism Overview
- Quiz & Worksheet - Neolithic Period Overview
- Quiz & Worksheet - When to Intercede in Conflict as a Manager
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- Anatomy & Physiology for Teachers: Professional Development
- Developing Employee Recognition Programs
- Western Civilization I: Help and Review
- History 106: The Civil War and Reconstruction
- CLEP Precalculus: Study Guide & Test Prep
- Studying for Chemistry 101
- PSAT Reading - About the Reading Section: Tutoring Solution
- Quiz & Worksheet - Characteristics of Generalized Anxiety Disorder
- Quiz & Worksheet - Items within a Merchandise Inventory
- Quiz & Worksheet - Non-Significant Outcomes in Psychology
- Quiz & Worksheet - Destruction Caused by Earthquakes
- Quiz & Worksheet - Chemical Classifications of Silicate Minerals

- Cellular Respiration: Biology Lab
- Reduction in Force: Definition & Guidelines
- Romeo and Juliet Act 4 Lesson Plan
- Does Your College GPA Matter?
- Getting Started with Study.com's College Courses: Student Tour
- Wisconsin Science Standards for 4th Grade
- Vietnam War During the Nixon Years: Learning Objectives & Activities
- Poetry Writing Prompts
- Georgia Science Standards for Kindergarten
- Homeschooling in Australia
- Minnesota Science Standards
- 4th Grade Science Projects

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject