What is Radioactive Decay? Half Life | Decay Constant | Activity (+ Problems Solving)

Introduction to Radioactive Decay Law

In this section, the speaker introduces the concept of radioactive decay and its different processes. They highlight the common mathematical expression that governs all radioactive decay processes.

Understanding Radioactive Decay

• Radioactive decay is a spontaneous process where a radioactive sample undergoes decay.

• Different types of radioactive decay include alpha, beta, and gamma decay.

• The mathematical expression that describes how the number of particles changes with time is known as the radioactive decay law.

Experimental Observation on Decay Rate

• The rate at which a substance decays is directly proportional to the number of particles present at any given time.

• The decay rate (DN/DT) represents the number of particles that disintegrate over a certain period of time.

• The proportionality constant in the radioactive decay law is called the decay constant or integration constant (lambda).

Deriving the Radioactive Decay Law

• By integrating the expression for DN/N = -lambda * DT, we can derive the equation for radioactive decay.

• At time T=0, if a container contains N₀ radioactive atoms, after some time T, it will have N remaining atoms.

• Using logarithmic simplification, we obtain N = N₀ * e^(-lambda * T), which represents exponential decay.

Characteristics of Decay Constant

This section focuses on understanding the characteristics and significance of the decay constant in relation to temperature, pressure, density, and elements undergoing radioactive decay.

Significance of Decay Constant

• The decay constant (lambda) determines how fast or slow a radioactive sample decays.

• It is not dependent on temperature, pressure, or density but rather specific to each decaying element.

Exponential Decay Graph

• A graph representing exponential decay shows the gradual decrease in the number of radioactive particles over time.

• The decay constant influences the rate of decay, with a larger lambda resulting in faster decay and vice versa.

Conclusion

The speaker concludes by summarizing the key points discussed, including the exponential nature of radioactive decay and the significance of the decay constant.

Key Takeaways

• Radioactive decay follows an exponential decay law.

• The number of particles remaining after time T can be calculated using N = N₀ * e^(-lambda * T).

• The decay constant determines the rate of decay and is specific to each decaying element.

Half-Life of Radioactive Samples

In this section, the concept of half-life and its relationship with radioactive decay are discussed.

Definition of Half-Life

• The half-life is the time period in which a sample decreases to half its original number.

• If a sample initially has n particles and it becomes n/2 after a certain time period T, then T is the half-life of that sample.

Relationship with Decay Law

• Applying the definition of half-life to the radioactive decay law (n = n₀e^(-λT)), where λ is the decay constant:

• When T is the half-life, n = n₀/2, and we have n/2 = n₀e^(-λT).

• Simplifying this equation gives us e^(-λT₁/₂) = 1/2 or λT₁/₂ = ln(2).

Relationship between Half-Life and Decay Constant

• The half-life (T₁/₂) can be calculated as ln(2)/λ.

• This relationship shows that if the decay constant is large, the decay happens rapidly, resulting in a shorter half-life. Conversely, if the decay constant is small, the decay occurs slowly, leading to a longer half-life.

Mean Life of Radioactive Samples

This section explores the concept of mean life and its relationship with radioactive samples.

Definition of Mean Life

• Mean life refers to the average lifespan or average time before an atom undergoes nuclear decay.

• Each individual atom within a sample may have different lifespans.

Calculation of Mean Life

• The mean life (τ) can be calculated by dividing the total lifespan of all atoms by the total number of particles present.

• Mathematically, mean life is represented as τ = 1/λ, where λ is the decay constant.

Relationship between Mean Life and Half-Life

• The half-life (T₁/₂) can be expressed as T₁/₂ = τ * 0.69.

• This relationship shows that the mean life is inversely proportional to the decay constant. A larger decay constant results in a shorter mean life, while a smaller decay constant leads to a longer mean life.

Summary

This section summarizes the key concepts discussed in the transcript.

• The half-life of a radioactive sample is the time it takes for the sample to decrease to half its original number of particles.

• The relationship between half-life and decay constant is given by T₁/₂ = ln(2)/λ.

• Mean life refers to the average lifespan of atoms within a radioactive sample.

• Mean life (τ) is inversely proportional to the decay constant (λ), with τ = 1/λ.

• The relationship between mean life and half-life is expressed as T₁/₂ = τ * 0.69.

The transcript provided information on specific calculations and relationships related to radioactive samples.

Radioactive Decay and Activity

In this section, we learn about the rate of radioactive decay and how it is related to the number of particles present. We also explore the concept of activity, which measures the rate at which a sample undergoes radioactive decay.

Rate of Decay and Number of Particles

• The rate at which radioactive decay occurs is directly proportional to the number of particles present.

• A large number of particles leads to a higher decay rate, while a small number results in a lower decay rate.

Decay Law with Respect to Activity

• The activity of a sample refers to the rate at which it undergoes radioactive decay.

• The activity can be expressed as R (symbol for activity) = -λN, where λ is the decay constant and N is the number of particles.

• The negative sign accounts for the decrease in particle count over time.

Units of Activity

• The unit for activity is Becquerel (Bq), defined as one disintegration per second.

• Another unit used is Curie (Ci), equal to 3.7 x 10^10 disintegrations per second.

Decay Law for Activity

• The relationship between activity (R) and time (T) can be described by R = R₀e^(-λT).

• R₀ represents the initial activity at T = 0, λ is the decay constant, and e is Euler's number.

Carbon Dating

Carbon dating is an application of radioactive decay laws used to determine the age of fossils. We solve a problem involving carbon dating using the decay law associated with activity.

Problem: Radioactive Carbon Dating

• Given a sample of carbon-14 with a half-life of 5730 years and a decay rate of 14 disintegrations per minute per gram of natural carbon.

• A fossil has a radioactivity of 4 disintegrations per minute per gram of its present carbon.

• We need to determine the age of the fossil.

Decay Law for Activity

• The decay law for activity is R = R₀e^(-λT), where R is the current activity, R₀ is the initial activity, λ is the decay constant, and T is time.

Solving the Problem

• Initially, R₀ = 14 and R = 4. We can substitute these values into the decay law equation.

• Simplifying, we find e^(-λT) = 4/14.

• Using the expression for half-life (0.69/λ), we can rewrite this as (0.69/(5730 years)) * T = ln(14/4).

• Solving for T, we find that approximately 10,358 years have elapsed since the fossil's formation.

Time Period for Thorium Disintegration

We explore another problem related to radioactive decay involving thorium. By applying the radioactive decay law associated with numbers, we calculate the time period required for a certain percentage of thorium to disintegrate.

Problem: Thorium Disintegration

• Given that thorium has a half-life of 1.4 x 10^10 years.

• We need to calculate the time period required for n percent of thorium to disintegrate.

Decay Law for Numbers

• The number of thorium particles at a given time can be expressed as N = N₀e^(-λT), where N is the current number, N₀ is the initial number, λ is the decay constant, and T is time.

Solving the Problem

• Initially, N₀ = 100% and after some time, N = n%. We can substitute these values into the decay law equation.

• Simplifying, we find e^(-λT) = n%/100%.

• Using the expression for half-life (0.69/λ), we can rewrite this as (0.69/(1.4 x 10^10 years)) * T = ln(n%/100%).

• Solving for T gives us the time period required for n percent of thorium to disintegrate.

The specific value of n is not provided in the transcript.

New Section

In this section, the speaker discusses the expression for radioactive decay and how to calculate the decay constant.

Expression for Radioactive Decay

• The speaker puts values in an expression and obtains 90/100 as a result.

• By rearranging the equation, they find that e to the power of minus lambda T is equal to n/9.

• Simplifying further, they get lambda T = ln(n/9).

New Section

This section focuses on converting half-life into decay constant.

Converting Half-Life to Decay Constant

• The speaker explains that lambda T can be written as 0.69 divided by the half-life multiplied by the time period.

• They equate this expression to ln(10/9) and solve for lambda T.

New Section

Here, the speaker discusses the time period it takes for 10% of a substance to disintegrate or for its total amount to reach 90%.

Time Period Calculation

• The time period is equal to the half-life (1.4 x 10^10 years) divided by 0.69 multiplied by ln(M/9).

• The result is computed as 2.1 x n^9.

New Section

This section emphasizes that understanding the decay law and related aspects makes problems of radioactivity simple.

Understanding Radioactivity Problems

• Problems related to radioactivity become simpler when one understands the decay law, including concepts like decay constant, half-life, mean life, and activity.

New Section

In this final section, the speaker concludes their discussion on radioactive decay law and encourages viewers to solve related problems.

Conclusion and Problem Solving

• The speaker concludes the video by summarizing the content covered on the radioactive decay law.

• They express hope that viewers have learned something and will be able to solve problems related to this topic.