Calculating Remaining Sample Mass

Oct 13, 2025 by windailys.com 40 views

Understanding Lutetium-176 Half-Life

Lutetium-176 is a radioactive isotope with a half-life of $$3.85 \times 10^{10}$$ years. This means that every $$3.85 \times 10^{10}$$ years, the amount of lutetium-176 will decrease by half. To understand the concept of half-life, imagine a sample of lutetium-176 with an initial mass of 16.8 g. After one half-life period, the mass will decrease to 8.4 g, and after another half-life period, it will decrease to 4.2 g, and so on.

Calculating Remaining Sample Mass

To calculate the remaining sample mass after a certain period, we need to determine how many half-life periods have passed. In this case, we are given a time period of $$1.155 \times 10^{11}$$ years. To find the number of half-life periods, we can divide the given time period by the half-life of lutetium-176.

import math

# Given values
half_life = 3.85 * (10 ** 10)  # years
time_period = 1.155 * (10 ** 11)  # years

# Calculate the number of half-life periods
num_half_lives = time_period / half_life

print(num_half_lives)

Running this code will give us the number of half-life periods that have passed. However, since we are dealing with a radioactive isotope, we need to consider the concept of decay. After each half-life period, the amount of lutetium-176 decreases by half. Therefore, we can use the formula for exponential decay to calculate the remaining sample mass.

Exponential Decay Formula

The formula for exponential decay is given by:

A(t) = A_0 \times (1/2)^{t/h}

where:

$A(t)$ is the amount of substance remaining at time $t$
$A_0$ is the initial amount of substance
$t$ is the time period
$h$ is the half-life of the substance

In this case, we can plug in the values as follows:

A(t) = 16.8 \times (1/2)^{1.155 \times 10^{11} / 3.85 \times 10^{10}}

Calculating Remaining Sample Mass

Now, let's calculate the remaining sample mass using the exponential decay formula.

import math

# Given values
initial_mass = 16.8  # g
half_life = 3.85 * (10 ** 10)  # years
time_period = 1.155 * (10 ** 11)  # years

# Calculate the remaining sample mass
remaining_mass = initial_mass * (1/2)**(time_period / half_life)

print(remaining_mass)

Running this code will give us the remaining sample mass after the given time period.

Conclusion

In conclusion, we have calculated the remaining sample mass of lutetium-176 after a time period of $$1.155 \times 10^{11}$$ years. Using the exponential decay formula, we found that the remaining sample mass is approximately 2.10 g.

Answer

Q: What is the half-life of lutetium-176?

A: The half-life of lutetium-176 is $$3.85 \times 10^{10}$$ years.

Q: How is the half-life of a radioactive isotope defined?

A: The half-life of a radioactive isotope is the time it takes for half of the initial amount of the isotope to decay.

Q: What is the formula for exponential decay?

A: The formula for exponential decay is given by:

A(t) = A_0 \times (1/2)^{t/h}

where:

$A(t)$ is the amount of substance remaining at time $t$
$A_0$ is the initial amount of substance
$t$ is the time period
$h$ is the half-life of the substance

Q: How do you calculate the remaining sample mass of a radioactive isotope?

A: To calculate the remaining sample mass of a radioactive isotope, you can use the exponential decay formula. You need to plug in the values of the initial amount of the isotope, the time period, and the half-life of the isotope.

Q: What is the significance of the half-life of a radioactive isotope?

A: The half-life of a radioactive isotope is significant because it determines how long it takes for the isotope to decay. This is important in various fields such as nuclear physics, chemistry, and medicine.

Q: Can you give an example of how to calculate the remaining sample mass of lutetium-176?

A: Yes, let's say we have an initial sample of 16.8 g of lutetium-176 and we want to calculate the remaining sample mass after a time period of $$1.155 \times 10^{11}$$ years. We can use the exponential decay formula as follows:

A(t) = 16.8 \times (1/2)^{1.155 \times 10^{11} / 3.85 \times 10^{10}}

Q: What is the remaining sample mass of lutetium-176 after a time period of $$1.155 \times 10^{11}$$ years?

A: Using the exponential decay formula, we can calculate the remaining sample mass of lutetium-176 after a time period of $$1.155 \times 10^{11}$$ years. The remaining sample mass is approximately 2.10 g.

Q: What is the correct answer to the problem?

A: The correct answer is A. 2.10 g.

Q: What is the significance of this problem?

A: This problem is significant because it demonstrates how to calculate the remaining sample mass of a radioactive isotope using the exponential decay formula. This is an important concept in various fields such as nuclear physics, chemistry, and medicine.

Q: Can you provide a Python code snippet to calculate the remaining sample mass of lutetium-176?

A: Yes, here is a Python code snippet to calculate the remaining sample mass of lutetium-176:

import math

# Given values
initial_mass = 16.8  # g
half_life = 3.85 * (10 ** 10)  # years
time_period = 1.155 * (10 ** 11)  # years

# Calculate the remaining sample mass
remaining_mass = initial_mass * (1/2)**(time_period / half_life)

print(remaining_mass)

Running this code will give us the remaining sample mass of lutetium-176 after a time period of $$1.155 \times 10^{11}$$ years.