Answer :
To solve this problem, we need to understand the concept of radioactive decay and the idea of a half-life.
1. Understanding Exponential Decay and Half-life:
- Exponential decay describes the process by which a substance decreases at a rate proportional to its current amount.
- The half-life is the time it takes for half of the substance to decay or reduce to half its initial amount.
2. Exponential Decay Formula:
- The general formula for exponential decay is [tex]\( N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{\text{half}}}} \)[/tex], where:
- [tex]\( N(t) \)[/tex] is the amount remaining at time [tex]\( t \)[/tex],
- [tex]\( N_0 \)[/tex] is the initial amount,
- [tex]\( T_{\text{half}} \)[/tex] is the half-life of the substance.
3. Decay Constant:
- The decay constant, often denoted by [tex]\( \lambda \)[/tex], is a parameter that describes the rate of decay. It can be calculated using the formula: [tex]\( \lambda = \frac{\ln(2)}{T_{\text{half}}} \)[/tex], where [tex]\( \ln(2) \)[/tex] is the natural logarithm of 2.
4. Application to the Given Problem:
- We are given the initial amount [tex]\( N_0 = 39.3 \)[/tex] grams and the half-life [tex]\( T_{\text{half}} = 17 \)[/tex] hours.
- Using these, we can compute the decay constant using the formula mentioned above.
5. Calculation Result:
- The decay constant [tex]\( \lambda \)[/tex] is calculated to be approximately 0.040773 per hour.
This decay constant helps us determine how quickly the radioactive substance decays over time. Without additional information such as a specific time or amount left, this is as far as we can analyze the problem for now.
1. Understanding Exponential Decay and Half-life:
- Exponential decay describes the process by which a substance decreases at a rate proportional to its current amount.
- The half-life is the time it takes for half of the substance to decay or reduce to half its initial amount.
2. Exponential Decay Formula:
- The general formula for exponential decay is [tex]\( N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{\text{half}}}} \)[/tex], where:
- [tex]\( N(t) \)[/tex] is the amount remaining at time [tex]\( t \)[/tex],
- [tex]\( N_0 \)[/tex] is the initial amount,
- [tex]\( T_{\text{half}} \)[/tex] is the half-life of the substance.
3. Decay Constant:
- The decay constant, often denoted by [tex]\( \lambda \)[/tex], is a parameter that describes the rate of decay. It can be calculated using the formula: [tex]\( \lambda = \frac{\ln(2)}{T_{\text{half}}} \)[/tex], where [tex]\( \ln(2) \)[/tex] is the natural logarithm of 2.
4. Application to the Given Problem:
- We are given the initial amount [tex]\( N_0 = 39.3 \)[/tex] grams and the half-life [tex]\( T_{\text{half}} = 17 \)[/tex] hours.
- Using these, we can compute the decay constant using the formula mentioned above.
5. Calculation Result:
- The decay constant [tex]\( \lambda \)[/tex] is calculated to be approximately 0.040773 per hour.
This decay constant helps us determine how quickly the radioactive substance decays over time. Without additional information such as a specific time or amount left, this is as far as we can analyze the problem for now.