An Element With A Mass Of 330 Grams Decays By 25.5% Per Minute. How Much Of The Element Remains After 5 Minutes, To The Nearest Tenth Of A Gram?

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Understanding Radioactive Decay

Radioactive decay is a process in which unstable atoms lose energy through radiation. This process is characterized by a decrease in the number of unstable atoms over time, resulting in a decrease in the mass of the substance. The rate of decay is often expressed as a percentage of the initial mass that decays per unit of time.

Calculating the Decay Rate

In this problem, we are given that the element decays by 25.5% per minute. This means that 25.5% of the initial mass of 330 grams is lost per minute. To calculate the remaining mass after 5 minutes, we need to calculate the mass that remains after each minute and then add up the remaining masses.

Calculating the Remaining Mass after 1 Minute

To calculate the remaining mass after 1 minute, we need to subtract the mass that decays from the initial mass. The mass that decays is 25.5% of 330 grams, which is:

mass_decayed = 330 * 0.255
mass_decayed = 84.15 grams

The remaining mass after 1 minute is:

mass_remaining = 330 - mass_decayed
mass_remaining = 245.85 grams

Calculating the Remaining Mass after 2 Minutes

To calculate the remaining mass after 2 minutes, we need to subtract the mass that decays from the remaining mass after 1 minute. The mass that decays is 25.5% of 245.85 grams, which is:

mass_decayed = 245.85 * 0.255
mass_decayed = 62.73 grams

The remaining mass after 2 minutes is:

mass_remaining = 245.85 - mass_decayed
mass_remaining = 183.12 grams

Calculating the Remaining Mass after 3 Minutes

To calculate the remaining mass after 3 minutes, we need to subtract the mass that decays from the remaining mass after 2 minutes. The mass that decays is 25.5% of 183.12 grams, which is:

mass_decayed = 183.12 * 0.255
mass_decayed = 46.73 grams

The remaining mass after 3 minutes is:

mass_remaining = 183.12 - mass_decayed
mass_remaining = 136.39 grams

Calculating the Remaining Mass after 4 Minutes

To calculate the remaining mass after 4 minutes, we need to subtract the mass that decays from the remaining mass after 3 minutes. The mass that decays is 25.5% of 136.39 grams, which is:

mass_decayed = 136.39 * 0.255
mass_decayed = 34.83 grams

The remaining mass after 4 minutes is:

mass_remaining = 136.39 - mass_decayed
mass_remaining = 101.56 grams

Calculating the Remaining Mass after 5 Minutes

To calculate the remaining mass after 5 minutes, we need to subtract the mass that decays from the remaining mass after 4 minutes. The mass that decays is 25.5% of 101.56 grams, which is:

mass_decayed = 101.56 * 0.255
mass_decayed = 25.89 grams

The remaining mass after 5 minutes is:

mass_remaining = 101.56 - mass_decayed
mass_remaining = 75.67 grams

Conclusion

To the nearest tenth of a gram, the element remains after 5 minutes is 75.7 grams.

Formula for Radioactive Decay

The formula for radioactive decay is:

A(t) = A0 * (1 - r)^t

Where: A(t) = mass remaining after time t A0 = initial mass r = decay rate t = time

In this problem, we can use the formula to calculate the remaining mass after 5 minutes:

A(5) = 330 * (1 - 0.255)^5 A(5) = 75.67 grams

This confirms the result we obtained using the step-by-step method.

References

Understanding Radioactive Decay

Radioactive decay is a process in which unstable atoms lose energy through radiation. This process is characterized by a decrease in the number of unstable atoms over time, resulting in a decrease in the mass of the substance. The rate of decay is often expressed as a percentage of the initial mass that decays per unit of time.

Q&A: Radioactive Decay

Q: What is the formula for radioactive decay?

A: The formula for radioactive decay is:

A(t) = A0 * (1 - r)^t

Where: A(t) = mass remaining after time t A0 = initial mass r = decay rate t = time

Q: What is the difference between radioactive decay and exponential decay?

A: Radioactive decay is a specific type of exponential decay that occurs in unstable atoms. Exponential decay is a more general term that refers to any process in which the rate of change is proportional to the current value.

Q: How do you calculate the remaining mass after a certain time using the formula for radioactive decay?

A: To calculate the remaining mass after a certain time using the formula for radioactive decay, you need to plug in the values for the initial mass, decay rate, and time. For example, if the initial mass is 330 grams, the decay rate is 25.5%, and the time is 5 minutes, the formula would be:

A(5) = 330 * (1 - 0.255)^5 A(5) = 75.67 grams

Q: What is the significance of the half-life in radioactive decay?

A: The half-life is the time it takes for half of the initial mass to decay. It is a measure of the rate of decay and is used to calculate the remaining mass after a certain time.

Q: Can you give an example of how to calculate the remaining mass after a certain time using the half-life?

A: Yes, let's say the half-life of a substance is 10 minutes and the initial mass is 100 grams. To calculate the remaining mass after 20 minutes, you would use the formula:

A(20) = 100 * (1/2)^2 A(20) = 25 grams

Q: What is the relationship between the decay rate and the half-life?

A: The decay rate and the half-life are inversely proportional. This means that as the decay rate increases, the half-life decreases, and vice versa.

Q: Can you give an example of how to calculate the decay rate using the half-life?

A: Yes, let's say the half-life of a substance is 10 minutes and the initial mass is 100 grams. To calculate the decay rate, you would use the formula:

r = 1 - (1/2)^(1/t) r = 1 - (1/2)^(1/10) r = 0.0952 or 9.52%

Q: What is the significance of the exponential decay in real-world applications?

A: Exponential decay is used in a variety of real-world applications, including:

  • Modeling population growth and decline
  • Predicting the spread of diseases
  • Calculating the remaining value of an investment
  • Determining the shelf life of products

Conclusion

Radioactive decay and exponential decay are important concepts in physics and mathematics. Understanding these concepts can help you calculate the remaining mass after a certain time and make predictions about real-world phenomena.

References