Question

1. iod - 131 has a half - life of 8.1 hours. how many grams remain after 40.5 hours if you start with 10.09 g?
2. the half - life of po - 214 is 2.4 minutes. if you start with 100.0 g of po - 214, how many grams would be left after 7.2 minutes has elapsed?
3. pu - 100 has a half - life of 4.00 days. if one had 6.02×10²³ atoms to start, how many atoms would be present after 20.0 days?
4. the half - life of uranium - 235 is 4.5 billion years. how much of a 13 g sample would be left after 13.5 billion years.

the half - life of \pennyium\
introduction: all radioactive matter decays. radioactive elements become nonradioactive over time. each radioactive element has a unique rate of decay. this \half - life\ is the average period of time it takes for half of the atoms in a radioactive sample to change into new atoms. this lab will give you a model to learn more about this concept.
procedure:

1. place ______ pennies in the box provided with the head sides up. the pennies will represent atoms of the hypothetical radioactive element \pennyium\.
2. cover the box and shake it (gently) for 3 seconds. this is one time interval.
3. remove the lid and take out any pennies that are heads side down (tails). these represent the atoms that decayed into a non - radioactive element.
4. record the numbers of decayed and remaining pennies (\atoms\) in your data table.
5. repeat steps 2 - 4 until all of the pennies have decayed.
6. make a graph of your data plotting the average number of radioactive atoms remaining versus time. consider all parts of a complete graph.

Explanation:

1. Question 1:

• Explanation:
• Step 1: Identify the half - life formula
• The formula for radioactive decay is $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $N$ is the final amount, $N_0$ is the initial amount, $t$ is the time elapsed, and $T_{1/2}$ is the half - life.
• Step 2: Analyze the problem
• For the first problem about iodine - 131 with $T_{1/2}=8.1$ hours and $t = 10.09$ hours, $N_0$ is not given, but we can still use the ratio concept. First, find the number of half - lives $n=\frac{t}{T_{1/2}}=\frac{10.09}{8.1}\approx1.246$. Then $N = N_0(\frac{1}{2})^{1.246}$.
• Step 3: Calculate the remaining fraction
• $(\frac{1}{2})^{1.246}\approx0.42$.
• Answer: Approximately 0.42 of the initial amount remains.

1. Question 2:

• Explanation:
• Step 1: Identify values
• Given $T_{1/2}=2.4$ minutes and $t = 7.2$ minutes. Calculate the number of half - lives $n=\frac{t}{T_{1/2}}=\frac{7.2}{2.4}=3$.
• Step 2: Use the decay formula
• Using $N = N_0(\frac{1}{2})^n$, with $N_0 = 100.0$ g and $n = 3$, we have $N=100.0\times(\frac{1}{2})^3$.
• Step 3: Calculate the remaining amount
• $N = 100.0\times\frac{1}{8}=12.5$ g.
• Answer: 12.5 g

1. Question 3:

• Explanation:
• Step 1: Identify values
• Given $T_{1/2}=4.00$ days, $N_0 = 6.02\times10^{23}$ atoms, and $t = 20.0$ days. Calculate the number of half - lives $n=\frac{t}{T_{1/2}}=\frac{20.0}{4.00}=5$.
• Step 2: Use the decay formula
• Using $N = N_0(\frac{1}{2})^n$, we substitute the values: $N=(6.02\times10^{23})\times(\frac{1}{2})^5$.
• Step 3: Calculate the remaining number of atoms
• $N=(6.02\times10^{23})\times\frac{1}{32}=1.88125\times10^{22}$ atoms.
• Answer: $1.88125\times10^{22}$ atoms

1. Question 4:

• Explanation:
• Step 1: Identify values
• Given $T_{1/2}=4.5$ billion years, $N_0 = 13$ g, and $t = 13.5$ billion years. Calculate the number of half - lives $n=\frac{t}{T_{1/2}}=\frac{13.5}{4.5}=3$.
• Step 2: Use the decay formula
• Using $N = N_0(\frac{1}{2})^n$, we substitute the values: $N = 13\times(\frac{1}{2})^3$.
• Step 3: Calculate the remaining amount
• $N=13\times\frac{1}{8}=1.625$ g.
• Answer: 1.625 g

Answer:

1. Question 1:

• Explanation:
• Step 1: Identify the half - life formula
• The formula for radioactive decay is $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $N$ is the final amount, $N_0$ is the initial amount, $t$ is the time elapsed, and $T_{1/2}$ is the half - life.
• Step 2: Analyze the problem
• For the first problem about iodine - 131 with $T_{1/2}=8.1$ hours and $t = 10.09$ hours, $N_0$ is not given, but we can still use the ratio concept. First, find the number of half - lives $n=\frac{t}{T_{1/2}}=\frac{10.09}{8.1}\approx1.246$. Then $N = N_0(\frac{1}{2})^{1.246}$.
• Step 3: Calculate the remaining fraction
• $(\frac{1}{2})^{1.246}\approx0.42$.
• Answer: Approximately 0.42 of the initial amount remains.

1. Question 2:

• Explanation:
• Step 1: Identify values
• Given $T_{1/2}=2.4$ minutes and $t = 7.2$ minutes. Calculate the number of half - lives $n=\frac{t}{T_{1/2}}=\frac{7.2}{2.4}=3$.
• Step 2: Use the decay formula
• Using $N = N_0(\frac{1}{2})^n$, with $N_0 = 100.0$ g and $n = 3$, we have $N=100.0\times(\frac{1}{2})^3$.
• Step 3: Calculate the remaining amount
• $N = 100.0\times\frac{1}{8}=12.5$ g.
• Answer: 12.5 g

1. Question 3:

• Explanation:
• Step 1: Identify values
• Given $T_{1/2}=4.00$ days, $N_0 = 6.02\times10^{23}$ atoms, and $t = 20.0$ days. Calculate the number of half - lives $n=\frac{t}{T_{1/2}}=\frac{20.0}{4.00}=5$.
• Step 2: Use the decay formula
• Using $N = N_0(\frac{1}{2})^n$, we substitute the values: $N=(6.02\times10^{23})\times(\frac{1}{2})^5$.
• Step 3: Calculate the remaining number of atoms
• $N=(6.02\times10^{23})\times\frac{1}{32}=1.88125\times10^{22}$ atoms.
• Answer: $1.88125\times10^{22}$ atoms

1. Question 4:

• Explanation:
• Step 1: Identify values
• Given $T_{1/2}=4.5$ billion years, $N_0 = 13$ g, and $t = 13.5$ billion years. Calculate the number of half - lives $n=\frac{t}{T_{1/2}}=\frac{13.5}{4.5}=3$.
• Step 2: Use the decay formula
• Using $N = N_0(\frac{1}{2})^n$, we substitute the values: $N = 13\times(\frac{1}{2})^3$.
• Step 3: Calculate the remaining amount
• $N=13\times\frac{1}{8}=1.625$ g.
• Answer: 1.625 g