Measurements on a certain isotope tell you that the decay rate decreases from 8318 8318 decays/min to 3091 3091 decays/min in 4.00 4.00 days. What is the half-life of this isotope?

Hi, everyone in this practice problem, we're being asked to calculate the half life, the health of iodine. We'll have a Geiger Mueller tube counting 650 disintegrations per second at a time, T equals to zero originating from an iodine sample. After 24 hours, it counts 597 disintegrations per second. And we're being asked to calculate the half life or T, half of iodine. The options given are a 22 hours. B one point oh nine days, C 8.15 days. And lastly D 11.6 days, the activity of the sample as a function of time is going to be given by the equation of A equals to a not multiplied by E to the power of negative lambda T or when we actually expand the negative lambda, this equation will then be equals to A not multiplied by E to the power of negative L N of two divided by T half multiplied by T. This will be equation one that we have here where a knot is going to be the activity at a time, T equals to zero. Which in this uh practice problem. A knot will be equals to 650 disintegrations per second. I'm just gonna in uh represent it with this per se for this in equations per second. And then the half is the half life. We wanna rearrange the equation one in order to make the half time the subject of the equation. So I'm gonna put the A K NOTT into the other side. So we will have the equation for a divided by A, no equals to E to the power of negative L N of two divided by T half multiplied by T. And then taking the L N for both sides, we can then get the equation of L N A divided by a knot equals two. So the L N of E will just then be whatever the power of E is. So the right side will just then be negative L N of two divided by C half multiplied by T just like so, and then in order for us to get T half, we wanna rearrange our equation so that T half will then be equal to negative L N R two multiplied by T. I'm gonna indicate that with the parenthesis to make it um more clearer for us divide everything with L N of A divided by A, not just like. So, so now we can actually uh input all of the information that we know. So let's start with L N of two negative L N of two multiplied by T, the T is going to be 24 hours, which is, uh, what is being asked in the problem statement, 24 hours, divide that with L N of A A is going to be 597 disintegrations per second. So that is the, um, count, uh, from the Geyger Mueller tube after 24 hours, which is why the T that we use is 24 hours and the A is 597 and then a knot or the original is disintegrations per seconds just like. So we can cross out the disintegrations per second and then calculating this, we will get our half five or T half to then be equal to 100 and 95.6 hours or um changing that into days that will equal to 8.15 days just like. So, so there we have it, the half life for the iodine is going to be 8.15 days and that will correspond to option C in our answer choices. So option C will be the answer to this particular practice problem and that'll be it for this video. If you guys still have any sort of confusion, please feel free to check out our other, lots of videos on similar topics available on our website. But other than that, that will be it for this one. Thank you.

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Problem 27

Textbook Question

Measurements on a certain isotope tell you that the decay rate decreases from $8318$ decays/min to $3091$ decays/min in $4.00$ days. What is the half-life of this isotope?

Verified step by step guidance

Step 1: Understand the concept of radioactive decay. The decay rate follows an exponential decay model, described by the equation:

N = N

₀e⁻λt, where

N

is the decay rate at time

t

N

₀ is the initial decay rate,

λ

is the decay constant, and

t

is the elapsed time.

Step 2: Use the given data to set up the equation. The initial decay rate is

8318

decays/min, the final decay rate is

3091

decays/min, and the elapsed time is

4.00

days. Convert the time into minutes:

4.00

days ×

1440

minutes/day =

5760

minutes.

Step 3: Rearrange the exponential decay formula to solve for the decay constant

λ

. Take the natural logarithm of both sides:

\ln (N / N

₀)=-λt. Substitute the values for

N

N

₀, and

t

Step 4: Once

λ

is calculated, use the relationship between the decay constant and the half-life:

T

_1/2=0.693/λ. Substitute the value of

λ

into this formula to find the half-life.

Step 5: Express the half-life in appropriate units (e.g., days or minutes) based on the context of the problem. Ensure the final answer is consistent with the given data and the units used throughout the solution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a characteristic rate for each isotope, often described by its decay constant, which is related to the probability of decay per unit time. Understanding this concept is crucial for analyzing how the number of undecayed nuclei decreases over time.

Half-Life

The half-life of a radioactive isotope is the time required for half of the radioactive nuclei in a sample to decay. It is a constant value unique to each isotope and is fundamental in calculating the remaining quantity of the substance after a given time. The half-life can be derived from the decay rate and is essential for understanding the longevity and stability of isotopes.

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, the number of undecayed nuclei decreases exponentially over time, which can be mathematically represented by the equation N(t) = N0 * e^(-λt), where N0 is the initial quantity, λ is the decay constant, and t is time. This concept is vital for calculating changes in decay rates over specified intervals.

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