To determine the average life (or mean lifetime) of a radioisotope given its decay constant, we can use the relationship between the decay constant (λ) and the average life (τ). The formula that relates these two quantities is: \[ \tau = \frac{1}{\lambda} \] ### Step-by-Step Explanation 1. **Identify the Decay Constant**: The decay constant (λ) is given as \(10^{-7} \, \text{s}^{-1}\). This means that the rate at which the radioisotope decays is \(10^{-7}\) per second. 2. **Apply the Formula**: To find the average life (τ), we substitute the value of λ into the formula: \[ \tau = \frac{1}{\lambda} = \frac{1}{10^{-7} \, \text{s}^{-1}} \] 3. **Calculate τ**: When we perform the calculation: \[ \tau = \frac{1}{10^{-7}} = 10^{7} \, \text{s} \] This means the average life of the radioisotope is \(1.00 \times 10^{7} \, \text{s}\). ### Conclusion The correct answer is **A. 1.00 x 10^7 s**. ### Explanation of Other Options - **Option B (6.93 x 10^6 s)**: This value is incorrect because it does not correspond to the calculation we performed. It seems to be a miscalculation or misunderstanding of the relationship between decay constant and average life. - **Option C (1.00 x 10^{-7} s)**: This option is incorrect as it represents the decay constant itself, not the average life. The average life is the inverse of the decay constant, not equal to it. - **Option D (6.93 x 10^7 s)**: This option is also incorrect. It is larger than the correct average life and does not relate to the decay constant provided. ### Common Pitfalls - **Confusing Decay Constant with Average Life**: It's crucial to remember that the decay constant (λ) and the average life (τ) are inversely related. Misunderstanding this relationship can lead to incorrect answers. - **Miscalculating the Inverse**: When calculating the average life, ensure that you correctly compute the inverse of the decay constant. A common mistake is to misinterpret the exponent when performing the calculation. ### Revision Summary - The average life (τ) of a radioisotope is calculated using the formula \(τ = \frac{1}{λ}\). - Given a decay constant of \(10^{-7} \, \text{s}^{-1}\), the average life is \(1.00 \times 10^{7} \, \text{s}\). - Be careful not to confuse the decay constant with the average life; they are inversely related. - Always double-check calculations to avoid common pitfalls in exponent handling.