An ancient wooden club is found that contains 270g of carbon and has an activity of 8.3 decays per second.
Determine its age assuming that in living trees the ratio of 14C/12C atoms is about 1.3x10^-12?
t= in years
The decay constant is
λ=ln2/T = 0.693/5730 = 1.209•10⁻⁴ yr⁻¹.
The fraction of ¹⁴C << ¹²C, => we use the atomic weight of ¹²C to find the number of carbon atoms in 270 g:
N= (270/12)•6.022•10²³ =1.35•10²⁵ atoms,
The number of ¹⁴C in the sample of living tree is
N₁₄=1.3•10⁻¹²•1.35•10²⁵ =1.76•10¹³ nuclei.
Decay Law
N=N₀•exp(-λt)
Activity
A= λN,
λN=λN₀•exp(-λt) (1 year= 3.16•10⁷ s),
8.3 = (1.209•10⁻⁴•1.76•10¹³/3.16•10⁷) exp(-1.209•10⁻⁴•t),
exp(-1.209•10⁻⁴•t) =
=8.3•3.16•10⁷/1.209•10⁻⁴•1.76•10¹³)=0.12,
-1.209•10⁻⁴•t = ln0.12 = - 2.095,
t=2.095/1.209•10⁻⁴ = 1.73•10⁴ yr.
Well, if we're talking about an ancient wooden club that contains carbon, I guess it's safe to say that it's a club with a lot of "clubbing" experience, huh? 😄 Now let's get down to the science!
To determine the age of this wooden club, we can use the radioactive decay of 14C. The decay rate is measured in decays per second, and the ratio of 14C/12C atoms in living trees is given as 1.3x10^-12. So, here's how we can calculate the age:
First, we need to find out the initial amount of 14C in the wooden club. Since we know the activity (8.3 decays per second), we can assume that initially, we had a lot more 14C. Ready for some math magic?
Let's multiply the activity by the half-life of 14C, which is approximately 5730 years. This will give us the initial number of 14C atoms:
Initial amount of 14C = activity * half-life = 8.3 decays/s * 5730 years.
Now, we can relate the initial amount of 14C to the current amount (270g). The ratio of 14C/12C atoms in the wooden club is about 1.3x10^-12. Dividing the current mass of 14C by this ratio will give us the initial mass of 14C:
Initial mass of 14C = 270g / (1.3x10^-12).
And finally, we can determine the age (t) in years using this equation:
t = -5730 years * ln(Initial mass of 14C / Initial amount of 14C).
Now, all we need to do is plug in the values and solve for t. Remember, this is all assuming that this wooden club didn't lie about its age on its dating profile! 🤣
To determine the age of the ancient wooden club, we can use the formula for radioactive decay:
N(t) = N(0) * e^(-kt)
Where:
N(t) is the number of radioactive atoms remaining at time t
N(0) is the initial number of radioactive atoms
k is the decay constant
t is the time in years
e is the mathematical constant approximately equal to 2.71828
First, let's calculate the initial number of radioactive atoms in the ancient wooden club.
We know that the ratio of 14C/12C atoms in living trees is about 1.3x10^-12. This means that for every 1.3x10^-12 14C atoms, there are 1 12C atoms.
So, we can calculate the number of 14C atoms in the ancient wooden club based on the amount of carbon. The molar mass of carbon is 12 g/mol, so we can use the following calculation:
Number of 14C atoms = (270g / 12g/mol) * (1.3x10^-12 14C atoms / 1 12C atom)
Now, let's substitute N(0) = Number of 14C atoms into the decay equation and solve for t.
N(t) = N(0) * e^(-kt)
We know that the activity (decays per second) is given as 8.3 decays per second. The activity is related to the number of radioactive atoms by the equation:
Activity = k * N(0)
We can rearrange this equation to solve for k:
k = Activity / N(0)
Now, we can substitute k and N(0) into the decay equation:
8.3 = (Number of 14C atoms) * e^(-k * t)
Now, we can solve for t:
t = -ln(8.3 / (Number of 14C atoms)) / k
Let's calculate the value of t using the given values.
To determine the age of the ancient wooden club, we can use the concept of radioactive decay and the known ratio of 14C/12C atoms.
The decay of 14C isotope follows an exponential decay model, and its activity can be measured in decays per second. The decay of 14C is also used for radiocarbon dating, as it has a known half-life of approximately 5730 years.
First, let's calculate the initial amount of 14C in the wooden club. We know that the wooden club contains 270g of carbon, and the ratio of 14C/12C atoms in living trees is approximately 1.3x10^-12.
Since the atomic mass of carbon-14 is 14 atomic mass units and the atomic mass of carbon-12 is 12 atomic mass units, the ratio of 14C/12C in the wooden club can be calculated as follows:
Ratio = (mass of 14C / atomic mass of 14C) / (mass of 12C / atomic mass of 12C)
Ratio = (270g / 14g) / (270g / 12g)
Ratio = (270 / 14) / (270 / 12)
Ratio = (270 * 12) / (14 * 270)
Ratio = 12 / 14
Ratio = 6 / 7
Now, we can establish the equation for the decay of 14C over time, t:
Current amount of 14C = Initial amount of 14C * e^(-λt)
In this equation, λ represents the decay constant, which is equal to ln(2) / half-life of 14C.
Using the known half-life of 5730 years, we can calculate the decay constant:
λ = ln(2) / 5730
Now, we can solve for t in the equation above by rearranging it:
t = -ln(Current amount of 14C / Initial amount of 14C) / λ
In our case, the current amount of 14C is given by the activity of the wooden club, which is 8.3 decays per second. The initial amount of 14C is obtained from the ratio of 14C/12C.
t = -ln(8.3 / (6/7)) / (ln(2) / 5730)
t = -ln(8.3 * (7/6)) / (ln(2) / 5730)
t = -ln(9.65) / (ln(2) / 5730)
t ≈ -2.267 / (0.693 / 5730)
t ≈ -2.267 / 0.693 * 5730
t ≈ -3274.17
Since time cannot be negative, we take the absolute value of the result:
t ≈ 3274.17 years
Therefore, the age of the ancient wooden club is approximately 3274 years.