archaeologist digging at the site of an ancient settlement discover the remains of a wooden structure. carbon 14 dating of the wood from the structure finds a decay rate of 180 decays/min. Knowing that the decay rate for living wood is 600 decays/min and the decay constant for 14C decay is 8270 years, determine the age of the structure. Show works!
Thank you!
To determine the age of the structure, we can use the concept of carbon dating. Carbon dating is based on the decay of carbon-14 (14C) isotopes in organic materials like wood. The decay rate of carbon-14 is measured in decays per minute (decays/min).
Let's break down the given information:
Decay rate of carbon-14 in living wood = 600 decays/min
Decay rate of carbon-14 in the wooden structure = 180 decays/min
Decay constant for carbon-14 decay (λ) = 8270 years
The decay rate of carbon-14 follows an exponential decay equation: R(t) = R₀ * e^(-λt), where R(t) is the current decay rate at time t, R₀ is the initial decay rate, and λ is the decay constant.
From the given information, we can set up two equations:
Equation 1: 600 = R₀ * e^(-λ * t_living)
Equation 2: 180 = R₀ * e^(-λ * t_structure)
We have two unknowns in these equations: t_living (age of living wood) and t_structure (age of the wooden structure). Our goal is to find t_structure.
To solve this problem, we need to eliminate R₀ from both equations. We can divide Equation 1 by Equation 2:
600 / 180 = (R₀ * e^(-λ * t_living)) / (R₀ * e^(-λ * t_structure))
3.33 = e^(-λ * t_living + λ * t_structure)
Taking the natural logarithm (ln) of both sides:
ln(3.33) = -λ * t_living + λ * t_structure
Now, we isolate t_structure:
ln(3.33) = λ * (t_structure - t_living)
t_structure - t_living = ln(3.33) / λ
t_structure = t_living + ln(3.33) / λ
We have t_living = 0, as it represents the age of living wood. Plugging in the values:
t_structure = 0 + ln(3.33) / 8270
t_structure = ln(3.33) / 8270 years
Using a calculator, we can find the approximate value of ln(3.33) = 1.2041. Substituting this value into the equation:
t_structure = 1.2041 / 8270 years
t_structure ≈ 0.0001457 years
Converting years to minutes:
t_structure ≈ 0.0001457 years * (365 days/year) * (24 hours/day) * (60 min/hour)
t_structure ≈ 0.126 minutes
Therefore, the age of the wooden structure is approximately 0.126 minutes.
I know it's cool to use no caps and no periods but it sure makes it hard to read when you don't help with proper sentence structure. (Well, you did use some.)
ln(No/N) = kt
No = 600
N = 180
k = 8270
t = ?