The half-life of sodium-24 is 14.9 hours. A hospital buys a 40 mg sample of sodium-24.
a) How many grams, to the nearest tenth, of sodium-21 will remain after 48 h?
b) After how long will only 2.5 mg remain?
I don't know how to write the exponential function in order to be able to solve this. ><
amount = 40(1/2)^(t/14.9) , where t is the number of hours.
test: if t = 0, Amount = 40 , Check!
if t = 14.9
amount = 40(1/2)^1 = 20 , Check!
a) if t = 48
Amount = 40(1/2)^(48/14.9)
= 40(1/2)^3.22147651
= 40(.107210899)
= appr. 4.29 g
b) 2.5 = 40(1/2)^(t/14.9)
.0625 = (.5)^(t/14.9)
t/14.9 = ln.0625/ln.5
t/14.9 = 4
t = 4(14.9) = 59.6 hours
To solve these problems, we will use the formula for exponential decay:
N(t) = N₀ * (1/2)^(t / t₁/₂)
Where:
N(t) is the amount of substance remaining at time t
N₀ is the initial amount of substance
t is the time that has passed
t₁/₂ is the half-life of the substance
Let's solve each problem step by step:
a) How many grams, to the nearest tenth, of sodium-21 will remain after 48 hours?
Given:
t = 48 hours
t₁/₂ = 14.9 hours
N₀ = 40 mg
First, convert N₀ from milligrams to grams:
N₀ = 40 mg = 0.04 g
Now, substitute the values into the formula:
N(t) = 0.04 g * (1/2)^(48 / 14.9)
Using a calculator, evaluate the expression:
N(t) ≈ 0.04 g * (0.197)
N(t) ≈ 0.00788 g
Therefore, after 48 hours, approximately 0.00788 grams of sodium-21 will remain.
b) After how long will only 2.5 mg remain?
Given:
N(t) = 2.5 mg
t₁/₂ = 14.9 hours
N₀ = 40 mg
Let's solve for t by rearranging the formula:
N(t) = N₀ * (1/2)^(t / t₁/₂)
Substitute the given values:
2.5 mg = 40 mg * (1/2)^(t / 14.9)
Divide both sides by 40 mg:
0.0625 = (1/2)^(t / 14.9)
Take the logarithm of both sides to solve for t:
log(0.0625) = log((1/2)^(t / 14.9))
Using the logarithm property log(a^b) = b * log(a):
log(0.0625) = (t / 14.9) * log(1/2)
Now, solve for t:
t / 14.9 = log(0.0625) / log(1/2)
Divide both sides by log(1/2):
t = (14.9 * log(0.0625)) / log(1/2)
Using a calculator, evaluate the expression:
t ≈ (14.9 * (-3.125)) / (-0.301)
t ≈ 154.3 hours
Therefore, it will take approximately 154.3 hours for only 2.5 mg of sodium-21 to remain.
No problem! I can help you understand how to write an exponential function and solve these problems involving half-life.
To start, let's define what a half-life is. In this case, the half-life of sodium-24 is 14.9 hours. This means that after 14.9 hours, half of the initial amount of sodium-24 will remain.
Now, let's solve each part of the problem step by step:
a) How many grams, to the nearest tenth, of sodium-24 will remain after 48 hours?
To answer this, we need to determine the number of half-lives that occur within 48 hours. We can do this by dividing the total time elapsed (48 hours) by the half-life of sodium-24 (14.9 hours):
number of half-lives = 48 hours / 14.9 hours ≈ 3.22
Since we can't have a fraction of a half-life, we round down to the nearest integer to get 3 half-lives.
Now, we can use the concept of exponential decay to calculate the remaining amount of sodium-24. The general formula for exponential decay is:
final amount = initial amount * (1/2)^(number of half-lives)
Given that the initial amount is 40 mg (or 0.04 g), the number of half-lives is 3, we can plug these values into the formula:
final amount = 0.04 g * (1/2)^3 ≈ 0.01 g
Therefore, after 48 hours, approximately 0.01 grams of sodium-24 will remain.
b) After how long will only 2.5 mg remain?
In this case, we need to determine the number of half-lives required for the initial amount of 40 mg to decay to 2.5 mg.
Using the same formula as before, but rearranging it to solve for the number of half-lives:
number of half-lives = log(base 1/2) (final amount / initial amount)
Plugging in the values, we get:
number of half-lives = log(base 1/2) (2.5 mg / 40 mg) ≈ 5.64
Again, we round down to the nearest integer, so 5 half-lives are required.
Finally, we can calculate the total time elapsed by multiplying the half-life by the number of half-lives:
total time = 14.9 hours * 5 ≈ 74.5 hours
Therefore, it will take approximately 74.5 hours for the initial amount of 40 mg to decay to 2.5 mg.
I hope this helps you understand how to write the exponential function and solve problems involving half-life! Let me know if you have any further questions.