Chapter 1: Problem 73
If the half-life of seaborgium- 266 is \(360 \mathrm{~ms}\), then \(k=(\ln (2)) / 360\).
Short Answer
Expert verified
The decay constant \( k \approx 0.001925 \text{ ms}^{-1} \).
Step by step solution
01
Understanding Half-life and Decay Constant
The half-life of a radioactive substance is the time needed for half of the substance to decay. The decay constant, denoted by \( k \), is related to the half-life \( t_{1/2} \) by the formula \( k = \frac{\ln(2)}{t_{1/2}} \).
02
Substituting Known Values
We are given that the half-life \( t_{1/2} \) of seaborgium-266 is \( 360 \text{ ms} \). Substitute this value into the formula: \[ k = \frac{\ln(2)}{360} \].
03
Calculating \( k \)
Calculate \( ln(2) \), which is approximately \( 0.693 \). Then substitute this into the equation: \[ k = \frac{0.693}{360} \].
04
Performing the Division
Perform the division: \[ k = \frac{0.693}{360} \approx 0.001925 \]. This is the decay constant \( k \).
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.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Half-life
The concept of half-life is central to understanding processes involving radioactive decay. Simply put, half-life is the time it takes for half of a radioactive substance to decay into another element or isotope. This means that after one half-life, only 50% of the original radioactive material remains. Half-life is a constant value for each radioactive isotope, meaning it remains the same regardless of the initial amount of substance you have.
One important property of half-life is that it defines how quickly a radioactive isotope will decay. For seaborgium-266, the half-life is extremely short, at only 360 milliseconds. This indicates a rapid decay process where the amount of seaborgium-266 decreases to half its original quantity in just over a third of a second.
Scientists and researchers use half-life to compare different radioactive elements and isotopes. When we understand the half-life, we can predict how long an isotope will remain active or how long it will emit radiation.
One important property of half-life is that it defines how quickly a radioactive isotope will decay. For seaborgium-266, the half-life is extremely short, at only 360 milliseconds. This indicates a rapid decay process where the amount of seaborgium-266 decreases to half its original quantity in just over a third of a second.
Scientists and researchers use half-life to compare different radioactive elements and isotopes. When we understand the half-life, we can predict how long an isotope will remain active or how long it will emit radiation.
Decay Constant
The decay constant, denoted as \( k \), is another essential concept related to radioactive decay. This constant provides a measure of the probability of decay of a single nucleus per unit time. It is inversely related to the half-life of a substance.
The relationship between half-life \( t_{1/2} \) and the decay constant is given by the formula:
\[ k = \frac{\ln(2)}{t_{1/2}} \]
This formula demonstrates how the decay constant can be calculated if the half-life is known. The term \( \ln(2) \) stems from the natural logarithm of 2, approximately equal to 0.693, which is a mathematical constant used in this specific relationship.
In the context of seaborgium-266, the decay constant tells us how frequently the isotope undergoes decay in any given time frame. A larger \( k \) value implies a shorter half-life and a faster decay rate, while a smaller \( k \) reflects a longer half-life and slower decay.
The relationship between half-life \( t_{1/2} \) and the decay constant is given by the formula:
\[ k = \frac{\ln(2)}{t_{1/2}} \]
This formula demonstrates how the decay constant can be calculated if the half-life is known. The term \( \ln(2) \) stems from the natural logarithm of 2, approximately equal to 0.693, which is a mathematical constant used in this specific relationship.
In the context of seaborgium-266, the decay constant tells us how frequently the isotope undergoes decay in any given time frame. A larger \( k \) value implies a shorter half-life and a faster decay rate, while a smaller \( k \) reflects a longer half-life and slower decay.
Radioactive Decay
Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. During this process, the isotope transforms into a different element or a more stable isotope of the same element. This happens because unstable atomic nuclei are constantly seeking to reach a more stable form.
There are several types of radioactive decay, including alpha decay, beta decay, and gamma decay. Each type involves the emission of different particles or energy, which are characteristic of specific elements and isotopes. The decay process continues until a stable nucleus is formed.
For example, in the case of seaborgium-266, it decays quickly because it possesses a half-life of only 360 milliseconds. This fast decay nature means that it quickly reduces in quantity and alters into stable nuclei or different elements. Understanding radioactive decay is important for applications in radiometric dating, nuclear medicine, and nuclear power generation.
There are several types of radioactive decay, including alpha decay, beta decay, and gamma decay. Each type involves the emission of different particles or energy, which are characteristic of specific elements and isotopes. The decay process continues until a stable nucleus is formed.
For example, in the case of seaborgium-266, it decays quickly because it possesses a half-life of only 360 milliseconds. This fast decay nature means that it quickly reduces in quantity and alters into stable nuclei or different elements. Understanding radioactive decay is important for applications in radiometric dating, nuclear medicine, and nuclear power generation.
Seaborgium-266
Seaborgium-266 is a synthetic element with a very short-lived isotope, meaning it does not occur naturally and can only be produced in a lab under specific conditions. Named after chemist Glenn T. Seaborg, it belongs to the group of transuranium elements which feature atomic numbers greater than uranium.
Seaborgium-266 is especially known for its extremely brief existence, with a half-life of merely 360 milliseconds. This characteristic makes it invaluable for research in the field of nuclear chemistry and advanced particle physics. Because its half-life is so short, it is primarily studied to understand nuclear reactions and the creation of superheavy elements.
The production of seaborgium-266 often involves colliding lighter elements at high speeds to form this heavier nucleus. Although its practical uses are limited due to its rapid decay, seaborgium-266's study provides precise insights into the forces at play within an atomic nucleus and contributes to the broader understanding of radioactive substances.
Seaborgium-266 is especially known for its extremely brief existence, with a half-life of merely 360 milliseconds. This characteristic makes it invaluable for research in the field of nuclear chemistry and advanced particle physics. Because its half-life is so short, it is primarily studied to understand nuclear reactions and the creation of superheavy elements.
The production of seaborgium-266 often involves colliding lighter elements at high speeds to form this heavier nucleus. Although its practical uses are limited due to its rapid decay, seaborgium-266's study provides precise insights into the forces at play within an atomic nucleus and contributes to the broader understanding of radioactive substances.
