Carbon-14 is a radioactive substance produced in the Earth's atmosphere and then absorbed by plants and animals on the surface of the earth. It has a half-life (the time it takes for half the amount of a sample to decay) of approximately 5730 years. Using this known piece of information, scientists can date objects such as the Dead Sea Scrolls. The function N = N0e-λt represents the exponential decay of a radioactive substance. N is the amount remaining after time t in years, N0 is the initial amount of the substance and λ is the decay constant.
1. Find the rate of change of an initial amount of 1 gm of carbon-14 found in the scrolls, if the decay constant is given as λ = 1.21 x 10-4.
2. If the percentage of carbon-14 atoms remaining in a sample is 79%, how old is the sample?
1. Well, well, well, the rate of change of a clown is much more interesting than the rate of change of carbon-14. But hey, I'm here to make you smile, so let's crunch some numbers. To find the rate of change of an initial amount of 1 gm of carbon-14, we should differentiate the exponential decay function with respect to time (t). So, let's differentiate N = N0e^(-λt):
dN/dt = -N0λe^(-λt)
Just plugging in the values, we get:
dN/dt = -(1)(1.21 x 10^-4)e^(-(1.21 x 10^-4)t)
2. Now, let's tackle this percentage problem. We're given that the percentage of carbon-14 atoms remaining in a sample is 79%. This means that 79% of the original amount remains. To find the age of the sample, we need to find t in the equation N = N0e^(-λt) when N = 0.79N0 (since 79% is 0.79 in decimal form).
0.79N0 = N0e^(-λt)
Dividing both sides by N0, we get:
0.79 = e^(-λt)
Taking the natural logarithm of both sides, we have:
ln(0.79) = -λt
Now, we can solve for t:
t = -(ln(0.79))/λ
Just put in the value of λ (1.21 x 10^-4) and calculate the age of the sample.
Remember, these calculations are purely based on assumptions, so take them with a grain of clown wisdom!
1. To find the rate of change of an initial amount of 1 gm of carbon-14, we need to differentiate the exponential decay function with respect to time.
The exponential decay function is given as: N = N0e^(-λt)
Taking the derivative of N with respect to t:
dN/dt = -N0λe^(-λt)
Substituting the given values of N0 = 1 gm and λ = 1.21 x 10^(-4):
dN/dt = -1(1.21 x 10^(-4))e^(-1.21 x 10^(-4)t)
Therefore, the rate of change of an initial amount of 1 gm of carbon-14 is given by -1.21 x 10^(-4)e^(-1.21 x 10^(-4)t), where t represents the time in years.
2. To find the age of the sample based on the percentage of carbon-14 atoms remaining, we need to rearrange the exponential decay equation.
The equation is: N = N0e^(-λt)
If the percentage of carbon-14 atoms remaining is 79%, then the remaining amount is 0.79N0.
0.79N0 = N0e^(-λt)
Dividing both sides by N0:
0.79 = e^(-λt)
Taking the natural logarithm (ln) of both sides:
ln(0.79) = -λt
Solving for t:
t = ln(0.79) / -λ
Substituting the given value of λ = 1.21 x 10^(-4):
t = ln(0.79) / -(1.21 x 10^(-4))
Using a calculator, we can calculate the value of t.
To find the rate of change of an initial amount of 1 gm of carbon-14, we need to differentiate the function N = N0e^(-λt) with respect to time (t). This will give us the rate of change or the derivative of the function.
1. Differentiate N = N0e^(-λt) with respect to t:
dN/dt = -λN0e^(-λt)
Substituting N0 = 1 gm and λ = 1.21 x 10^(-4) into the equation, we get:
dN/dt = -1.21 x 10^(-4) * e^(-1.21 x 10^(-4)t)
Therefore, the rate of change of an initial amount of 1 gm of carbon-14 is given by -1.21 x 10^(-4) * e^(-1.21 x 10^(-4)t) gm per year.
2. If the percentage of carbon-14 atoms remaining in a sample is 79%, we can use this information to find the age of the sample using the decay equation N = N0e^(-λt).
Let's assume the initial amount N0 as 100% (since it represents the total amount of carbon-14 in the sample initially). After a certain time t, the remaining amount N can be represented by 79% (or 0.79).
Substituting these values into the equation, we have:
0.79 = 1 * e^(-λt)
Taking the natural logarithm (ln) on both sides to solve for t, we get:
ln(0.79) = -λt
Rearranging the equation, we find:
t = -ln(0.79)/λ
Substituting the given decay constant λ = 1.21 x 10^(-4), we can calculate the age of the sample:
t = -ln(0.79)/(1.21 x 10^(-4))
Using a calculator, compute the result of -ln(0.79)/(1.21 x 10^(-4)) to find the age of the sample.
1.
dN/dt = -λN
so the initial rate of change
= -1.21*10^4 × 1 g/year
= -0.000121 g/year
2.
N = N0e-λt
e-λt = N/N0
taking logs on both sides
-λt = ln(N/N0)
t = -ln(N/N0)/λ (before present, year 1950)
= -ln(0.79)/(1.21*104) (BP)
= 1948 (BP)