Need help understanding this problem. Looks like an exponential growth/decay.
Magnesium-27 has a half life of 9.45 minutes and can be described by using the equation “y=ae^-kt .
What is the decay constant (k) for Magnesium-27?
How many mg of Magnesium-27 will remain after 1.5 hours from an initial amount of 1000 mg?
How long will it take for 1000 mg of Magnesium to decrease to 780 mg?
Any explanation would be greatly appreciated!!
the amount halves every 9.45 minutes, so
y = a(1/2)^(t/9.45)
but,1/2 = e^-0.693 so
y = a(e^-0.693)^(t/9.45) = ae^(-0.0733t)
where t is in minutes
Now use that to evaluate your other questions
Oh, magnesium-27 and its exponential decay. What a classic!
To find the decay constant (k) for Magnesium-27, we can use the half-life formula. Half-life represents the time it takes for half of the substance to decay. So, if the half-life is 9.45 minutes, we can set up the equation:
0.5 = e^(-k * 9.45)
Now we just need to solve for k. But hey, don't worry, I'm here to help!
Now, let's move on to the next question. We've got 1000 mg of Magnesium-27 and we want to know how much will remain after 1.5 hours. To find that out, we'll use our handy-dandy equation: y = ae^(-kt).
So, let's plug in the values we know: a = 1000 mg, t = 1.5 hours (which is 1.5 * 60 minutes). And don't forget, we've already figured out the value of k!
Lastly, you're curious about how long it will take for 1000 mg of Magnesium-27 to decrease to 780 mg. Well, once again, we can use our trusty equation: y = ae^(-kt). This time, we need to solve for t.
Let's plug in our values: a = 1000 mg, y = 780 mg, and k (which we found earlier).
Hope this explanation tickled your funny bone, even if exponential decay might not be the most exciting topic!
To find the decay constant (k) for Magnesium-27, we can use the half-life formula:
t_1/2 = ln(2) / k
where t_1/2 is the half-life of the substance. In this case, the half-life is given as 9.45 minutes. Plugging in the values:
9.45 = ln(2) / k
To isolate k, we multiply both sides of the equation by k:
9.45k = ln(2)
Now, divide both sides by 9.45:
k = ln(2) / 9.45
Using a calculator, approximate the value of k:
k ≈ 0.0737 (rounded to four decimal places)
Now let's move on to the second question.
To find how much Magnesium-27 will remain after 1.5 hours, we need to substitute the given values into the equation y = ae^(-kt).
Let's break down the given information:
- Initial amount (a) = 1000 mg
- Time (t) = 1.5 hours = 90 minutes
- Decay constant (k) = 0.0737 (from the previous calculation)
Now plug these values into the equation:
y = 1000e^(-0.0737 * 90)
Using a calculator, you can evaluate this expression:
y ≈ 43.75 mg (rounded to two decimal places)
Approximately 43.75 mg of Magnesium-27 will remain after 1.5 hours.
Finally, let's move on to the last question.
To find the time it takes for 1000 mg of Magnesium-27 to decrease to 780 mg, we need to rearrange the equation slightly:
780 = 1000e^(-0.0737t)
Now, divide both sides by 1000:
0.78 = e^(-0.0737t)
Next, take the natural logarithm of both sides:
ln(0.78) = -0.0737t
Now, divide both sides by -0.0737:
t ≈ -ln(0.78) / 0.0737
Using a calculator, you can evaluate this expression:
t ≈ 19.65 minutes (rounded to two decimal places)
So, it will take approximately 19.65 minutes for 1000 mg of Magnesium-27 to decrease to 780 mg.
To find the decay constant (k) for Magnesium-27, we can use the equation "y = ae^(-kt)" where:
- y is the amount of Magnesium-27 at a given time
- a is the initial amount of Magnesium-27
- e is the base of the natural logarithm (approximately 2.71828)
- k is the decay constant
- t is the time in the same units as the half-life
In this case, the half-life of Magnesium-27 is given as 9.45 minutes.
To find the decay constant (k), we can use the formula:
k = (ln(2)) / half-life
Substituting the given half-life into the formula, we have:
k = (ln(2)) / 9.45 minutes
Calculate this value to find the decay constant.
Now let's move on to the other questions.
1. To find how much Magnesium-27 will remain after 1.5 hours, we need to convert the time to minutes since the half-life is given in minutes.
1.5 hours = 1.5 x 60 minutes = 90 minutes
Using the equation y = ae^(-kt) with the values we have:
a = 1000 mg (initial amount)
t = 90 minutes
k = decay constant (calculated earlier)
Plug in these values into the equation and solve for y. This will give you the remaining amount of Magnesium-27 after 1.5 hours.
2. To find how long it will take for 1000 mg of Magnesium-27 to decrease to 780 mg, we need to solve for t in the equation y = ae^(-kt).
a = 1000 mg (initial amount)
y = 780 mg
k = decay constant (calculated earlier)
Plug in these values into the equation and solve for t. This will give you the time it takes for the amount to decrease from 1000 mg to 780 mg.
Remember to use the units consistently throughout the calculations (minutes for the half-life and time, milligrams for the amount of Magnesium-27).
Hope this explanation helps you understand and solve the problem!