4) You discover a new radioactive substance. You have 45 grams of the substance initially. After 4 hours you have 33.6 grams remaining. a) What is the half-life of this radioactive substance? b) What mass will remain after 14 hours? c) Determine the time required for there to be only 8 grams remaining.

To find the half-life of the new radioactive substance, we start with the formula for exponential decay, which states that the remaining mass \( M \) at time \( t \) is related to the initial mass \( M_0 \) and half-life \( t_{1/2} \). The equation can be expressed as: \[ M = M_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \] Given that 45 grams decays to 33.6 grams in 4 hours, we can plug in the values: \[ 33.6 = 45 \left(\frac{1}{2}\right)^{\frac{4}{t_{1/2}}} \] By solving for \( t_{1/2} \), we can rearrange this equation: \[ \left(\frac{1}{2}\right)^{\frac{4}{t_{1/2}}} = \frac{33.6}{45} = 0.748 \] We can take the logarithm of both sides to solve for \( t_{1/2} \): \[ \frac{4}{t_{1/2}} \log(0.5) = \log(0.748) \] After calculating, you'll find that the half-life \( t_{1/2} \) is approximately 2.0 hours. To determine the mass remaining after 14 hours, we can use the half-life found previously. The number of half-lives in 14 hours is: \[ \text{Number of half-lives} = \frac{14}{t_{1/2}} = \frac{14}{2} = 7 \] Each half-life reduces the mass by half, starting from 45 grams: \[ M = 45 \left(\frac{1}{2}\right)^{7} = 45 \cdot \frac{1}{128} \approx 0.3515625 \text{ grams} \] To find the time required for the mass to reduce to 8 grams, we can again use the decay formula: \[ 8 = 45 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \] Solving for \( t \): \[ \left(\frac{1}{2}\right)^{\frac{t}{2}} = \frac{8}{45} \] Taking the logarithm of both sides gives: \[ \frac{t}{2} \log(0.5) = \log\left(\frac{8}{45}\right) \] Thus, \[ t = 2 \cdot \frac{\log\left(\frac{8}{45}\right)}{\log(0.5)} \] Calculating this yields that you will need approximately 12 hours for the mass to decay to 8 grams. Now go ahead and impress your friends with your newfound radioactive decay knowledge at your next science gathering!