Question

The half-life of polonium-218 is 3.0 min. If you start with 20.0 g, how long will it be before only 1.0 g remains?

Short answer

It will take approximately 12.96 minutes for the sample of polonium-218 to decay from 20.0 g to 1.0 g.

Step-by-step solution

Identify the relevant variables and constants

Initial mass (m_initial) = 20.0 g Final mass (m_final) = 1.0 g Half-life (t_half) = 3.0 min

Use the half-life formula to determine the number of half-lives that have passed

The formula relating initial mass, final mass and the number of half-lives is: m_final = m_initial * (1/2)^n Where n is the number of half-lives passed. We need to solve for n: 1.0 = 20.0 * (1/2)^n

Solve for the number of half-lives (n)

We can solve the above equation for n: (1/2)^n = 1.0 / 20.0 We can use logarithms to calculate 'n', specifically using natural logarithm 'ln': n = ln(1.0 / 20.0) / ln(1/2) Now calculate n: n ≈ 4.32

Calculate the total time it takes for the sample to decay

Since we have the value of n (number of half-lives), we can now find the total time it took for the mass to decay from 20.0 g to 1.0 g: time = n * t_half time = 4.32 * 3.0 time ≈ 12.96 min So, it will take approximately 12.96 minutes for the sample of polonium-218 to decay from 20.0 g to 1.0 g.

Key concepts

Polonium-218

Polonium-218 is a naturally occurring radioactive isotope. It belongs to the polonium family of elements. What makes it particularly interesting is its short half-life of just three minutes. This means that Polonium-218 decays rapidly into other elements in a relatively short period. This rapid decay is an essential property for calculating how long it takes for a given amount of this substance to decrease to a specified quantity. Some fascinating facts about Polonium-218:

• It's part of the uranium-238 decay series, where uranium ultimately breaks down into lead through a chain of decay steps.
• Polonium-218 emits radioactive alpha particles during its decay process.
• It plays a minor role in determining the age of uranium-rich minerals.

This makes understanding its behavior important in fields such as geology and radioactive dating, where knowing the timespan of such decay processes is crucial.

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. There are different types of radioactive decay, including alpha decay, beta decay, and gamma decay. In the case of Polonium-218, it undergoes alpha decay. Here's how this process could be described:

• During alpha decay, Polonium-218 emits an alpha particle, which comprises 2 protons and 2 neutrons.
• As a result of emitting this alpha particle, the nucleus loses some of its mass, becoming a new element altogether.
• This transformation is part of a series of decays each element in this series undergoes, until a stable element, such as lead, is formed.

Radioactive decay is a random process. However, scientists can predict its behavior statistically using half-lives. The half-life is the time required for half of the radioactive atoms in a sample to decay. This concept helps us understand and calculate how materials like Polonium-218 change over time.

Natural Logarithm

Logarithms are mathematical tools used to simplify the handling of exponential relationships. The natural logarithm, denoted as ln, utilizes the base of 'e' (approximately 2.718) and is especially useful in continuous growth or decay situations.When dealing with radioactive decay and half-lives, the natural logarithm can help us solve for the number of half-lives (n) a substance has undergone. Here's the connection:

• The equation relating initial mass, final mass, and half-lives uses an exponential relationship: \( m_{final} = m_{initial} \cdot (1/2)^n \).
• To solve for the unknown 'n', the equation is rearranged using the natural logarithm: \( n = \frac{\ln(m_{final} / m_{initial})}{\ln(1/2)} \).
• This principle allows us to calculate 'n' and, subsequently, the time taken for a radioactive substance to decay by using its half-life.

Using natural logarithms in decay calculations simplifies the process because it transforms multiplicative decay into a linear form. This approach makes solving decay problems more straightforward and manageable.