Final answer:
The half-life of the unknown radioactive element cannot be calculated without the precise decay rate formula, but for a 100 mg sample to decay to 78 mg, it would take approximately 394.7 days.
Step-by-step explanation:
To determine the half-life of an unknown radioactive element, we use the provided information that the radioactivity of a sample decreases by 38 percent in 680 days. This means that in 680 days, the sample retains 62 percent (100% - 38%) of its initial radioactivity. To find the half-life, we need to find the time it takes for the sample to decrease to 50 percent of its original activity. As the decay follows first-order kinetics and the relationship is not given linearly, we must use a logarithmic approach or iterative methods to find the exact half-life. Without the precise decay rate formula, we cannot calculate the half-life.
To calculate how long it will take for a 100 mg sample to decay to 78 mg, we first consider that losing 22 mg corresponds to a 22 percent decrease in mass. If 38 percent decay occurs in 680 days, a proportional amount of time can be calculated for a 22 percent decrease. As such, (22 / 38) multiplied by 680 days gives us the number of days needed for a 22 percent decrease, which is approximately 394.7 days.