Question

Element \(X\) is radioactive and decays via \(\alpha\) decay with a half-life of four days. If 12.5 percent of an original sample of element \(x\) remains after \(n\) days, what is the value of \(n ?\) (A) 4 (B) 8 (C) 12 (D) 16

Short answer

n is 12 days, so the correct answer is (C).

Step-by-step solution

- Understand Half-life

Half-life is the time it takes for half of a sample of a radioactive element to decay. In this problem, the half-life of element X is given as 4 days.

- Determine remaining fraction

Since 12.5% (or 0.125) of the original sample remains, we need to determine how many half-lives (cycles of decay) result in this remaining fraction.

- Use the formula for decay

The remaining amount of a sample after n half-lives can be calculated using the formula: \[ \text{Remaining fraction} = \frac{1}{2^n} \] In this problem, \[ \frac{1}{2^n} = 0.125 \]

- Solve for n

To find n, solve the equation: \[ 0.125 = \frac{1}{2^n} \] We know that \[ 0.125 = \frac{1}{8} \] and since \[ 8 = 2^3 \], it follows that \( n = 3 \).

- Calculate total time in days

Each half-life cycle (n = 1) is 4 days. Since we have determined there are 3 half-lives, multiply these together to get the total time elapsed: \[ n \times \text{half-life} = 3 \times 4 = 12 \text{ days} \]

- Verify the answer

The value matches choice (C) in the problem statement. This confirms that after 12 days, 12.5% of the original sample remains.

Key concepts

Half-life

Half-life is a key concept in understanding radioactive decay. It is the time required for half of the atoms in a radioactive sample to decay. For example, if you start with 100 atoms of element X and its half-life is 4 days, after 4 days, you would have 50 atoms left. This process continues, so after another 4 days (8 days total), you would have 25 atoms. Over time, the number of undecayed atoms decreases exponentially. Half-life helps in predicting how quickly a radioactive substance will diminish. When solving problems involving half-life, always remember that the time it takes for each successive halving remains constant.

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable atom emits an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, exactly like a helium nucleus. This emission results in the atom losing 4 units of mass and 2 units of positive charge; it changes the parent atom to a different element. For instance, if element X undergoes alpha decay, it loses an alpha particle and transforms into a new element, often changing its chemical behavior. Understanding alpha decay is important to grasp how certain elements evolve over time and solve related problems.

Decay Formula

The decay formula is used to calculate the remaining amount of a radioactive substance over time. It's typically represented as:  \[ \text{Remaining Fraction} = \frac{1}{2^n} \] where 'n' is the number of half-life periods that have elapsed. This formula helps in determining how much of a substance remains after a certain number of half-lives. For example, if you start with 100 grams of element X and it has a half-life of 4 days, using the formula, after 8 days (2 half-lives), you would have: \[ \frac{1}{2^2} = \frac{1}{4}, \] so you’d have 25 grams left. By plugging in different values, you can predict how much of a substance remains at any given time.

Exponential Decay

Exponential decay describes how the quantity of a radioactive substance decreases over time in a predictable pattern. This type of decay occurs continuously and rapidly at first, then slows down. In mathematical terms, exponential decay can be explained using the formula: \[ N(t) = N_0 e^{-\text{λ}t}, \] where: \ * N(t) is the quantity remaining at time t, \ * N_0 is the initial quantity, \ * e is the base of the natural logarithm, \ * λ (lambda) is the decay constant, and \ * t is the time elapsed. While exponential decay may look daunting, the key takeaway is that it helps predict how quickly a sample diminishes over time. The decay constant λ can be related to the half-life with the formula: \[ \text{λ} = \frac{\text{ln}(2)}{\text{half-life}}. \] Understanding these relationships helps solve various problems involving radioactive decay predictably.