Type the correct answer in the box. Express your answer to three significant figures.
The half-life of carbon-14 is 5,730 years. Dating organic material by looking for [tex]$C -14$[/tex] can't be accurately done after 50,000 years. Suppose a fossilized tree branch originally contained 4.30 grams of C-14. How much C-14 would be left after 50,000 years? Use the formula [tex]$N=N_0\left(\frac{1}{2}\right)^{\dagger}$[/tex].
A tree branch that originally had 4.3 grams of carbon-14 will have $\square$ grams after 50,000 years.
Answer (2)
Calculate the exponent: † = 5730 50000 ≈ 8.726 .
Substitute the values into the formula: N = 4.30 ( 2 1 ) 8.726 .
Calculate the remaining amount: N ≈ 0.010 grams.
The amount of C-14 left after 50,000 years is 0.010 grams.
Explanation
Understanding the Problem We are given the formula for exponential decay of Carbon-14: N = N 0 ( 2 1 ) † , where N is the amount of C-14 remaining after time t , N 0 is the initial amount of C-14, and † = t 1/2 t where t 1/2 is the half-life of C-14. We are given N 0 = 4.30 grams, t = 50 , 000 years, and t 1/2 = 5 , 730 years. Our goal is to find N .
Calculating the Exponent First, we calculate the value of † : † = t 1/2 t = 5730 50000 ≈ 8.726 .
Substituting Values Now, we substitute the given values into the formula: N = 4.30 ( 2 1 ) 5730 50000 = 4.30 ( 2 1 ) 8.726 .
Calculating the Remaining Amount Calculating the remaining amount of C-14: N = 4.30 × ( 0.5 ) 8.726 ≈ 4.30 × 0.002325 ≈ 0.010 . Rounding to three significant figures, we get N = 0.010 grams.
Final Answer Therefore, after 50,000 years, there would be approximately 0.010 grams of C-14 left in the tree branch.
Examples
Carbon-14 dating is a real-world application of exponential decay, used to estimate the age of organic materials. For example, archaeologists use carbon-14 dating to determine the age of ancient artifacts, such as wooden tools or human remains. By measuring the amount of C-14 remaining in the sample and comparing it to the initial amount, scientists can estimate how long ago the organism died. This technique relies on the constant decay rate of C-14, with a half-life of 5,730 years, making it a valuable tool for understanding the past.
After 50,000 years, approximately 0.010 grams of Carbon-14 will remain in the tree branch. This is calculated using the formula for exponential decay based on the half-life of C-14. Understanding this decay process allows scientists to estimate the ages of organic materials accurately.
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