Question

Determine the number of half-lives that must pass for only $1\mathrm{\%}$ of a particular radioisotope to remain.

Short answer

The number of half-lives that must pass for only 1% of a particular radioisotope to remain is approximately 7. This calculation was derived using the half-life decay formula and solving for n, with the equation $n={\displaystyle \frac{\ln (0.01)}{\ln ({\displaystyle \frac{1}{2}})}}$, which gave a result of 6.64, and rounded up to the nearest whole number.

Step-by-step solution

Understand the concept of half-life

A half-life is the time it takes for half of the radioactive atoms in a sample to decay. In each half-life, the remaining amount of the radioisotope gets reduced by half. To find out the number of half-lives passed for only 1% of the radioisotope to remain, we will first set up an equation representing the remaining percentage after each half-life.

Set up the equation

Let $n$ be the number of half-lives that have passed. After each half-life, the remaining percentage of the radioisotope will be reduced by half. To represent this mathematically, we can use the equation: $Remaining\mathrm{\%}=Initial\mathrm{\%}\times ({\displaystyle \frac{1}{2}}{)}^n$ Initially, we have 100% (1 as a decimal) of the radioisotope remaining. We want only 1% (0.01 as a decimal) to remain: $0.01=1\times ({\displaystyle \frac{1}{2}}{)}^n$

Solve for the number of half-lives

Now, we have the equation $0.01=({\displaystyle \frac{1}{2}}{)}^n$. We need to solve for $n$. To do this, we can take the natural logarithm of both sides: $\ln (0.01)=\ln (({\displaystyle \frac{1}{2}}{)}^n)$ Using the logarithm property, we can move the exponent ($n$) to the front: $\ln (0.01)=n\times \ln ({\displaystyle \frac{1}{2}})$ Now, we can solve for $n$: $n={\displaystyle \frac{\ln (0.01)}{\ln ({\displaystyle \frac{1}{2}})}}$

Calculate the number of half-lives

Using a calculator to find the values of the logarithms: $n={\displaystyle \frac{\ln (0.01)}{\ln ({\displaystyle \frac{1}{2}})}}\approx 6.64$ As the number of half-lives must be a whole number, we round up to the nearest whole number: $n=7$ So, 7 half-lives must pass for only 1% of the radioisotope to remain.

Key concepts

Radioisotope Decay

Radioisotope decay is a fascinating process where unstable atomic nuclei lose energy by emitting radiation. This process transforms the nuclei into more stable forms. Over time, the atoms of the radioisotope decay and become slightly different elements or isotopes. Decay occurs at a predictable rate, which makes it useful for dating materials and understanding the age of objects. Each radioisotope has a characteristic time scale known as its half-life. During each half-life, half of the radioactive atoms will decay, leaving half undecayed. Understanding radioisotope decay is crucial for calculating how much of a substance will remain over certain periods.

Natural Logarithm

The natural logarithm, often denoted as $\ln$, is a logarithm to the base $e$, where $e\approx 2.71828$. It is commonly used in solving equations involving exponential decay, such as radioisotope decay problems. In our half-life calculations, we use the property of logarithms which allows us to simplify expressions. For example, the equation $\ln (a^b)=b\times \ln (a)$ is extremely helpful. By applying the natural logarithm in decay equations, we can solve for unknowns like the number of half-lives $n$. Ultimately, the application of natural logarithms grants us the ability to solve complex decay equations that involve exponents. It allows us to rearrange and better understand the relationships in decay processes.

Exponential Decay

Exponential decay is a process where the quantity of a substance decreases at a rate proportional to its current value. This is a defining characteristic of radioisotope decay. In mathematical terms, this is represented as $N(t)=N_0\times e^{-\lambda t}$, where $N(t)$ is the quantity remaining at time $t$, $N_0$ is the initial quantity, and $\lambda$ is the decay constant. For half-life calculations, we frequently use a simplified form: $N=N_0\times {\left(\frac{1}{2}\right)}^n$. Each half-life results in a 50% reduction of the initial quantity. This consistent halving is why its decay curve is steeped and continuous, providing a clear picture of the decay progress over time.

Radioactive Atoms

Radioactive atoms are the building blocks of radioisotopes, which are unstable variants of the standard atoms of a chemical element. These atoms possess excess nuclear energy, making them radioactive. Because they are unstable, they release energy in the form of radiation to transform into more stable configurations. This transformation can result in the emission of alpha particles, beta particles, or gamma rays. Over time, the proportion of radioactive atoms decreases as they decay. Understanding the behavior of radioactive atoms provides insights into broader scientific areas such as nuclear physics, medicine, and environmental science.