Question 1: Radioactive decay is a random process where unstable nuclei spontaneously emit particles to become more stable. The rate at which a radioactive substance decays is described by the decay equation N = N_0 e^{-λt}. Explain what each symbol in this equation represents.

Answer: N is the number of radioactive nuclei remaining after time t, N_0 is the initial number of nuclei at time zero, λ is the decay constant, and t is the time elapsed.

Question 2: Describe the mechanism by which unstable nuclei undergo radioactive decay.

Answer: Unstable nuclei spontaneously emit particles such as alpha or beta particles, resulting in a change to a more stable nucleus, often accompanied by energy release.

Question 1: A sample contains 1,000 radioactive nuclei. The decay constant (λ) for this isotope is 0.002 per day. Calculate the number of nuclei remaining after 100 days. Use the decay equation N = N_0 e^{-λt}.

Answer: N = 1000 × e^{-0.002 × 100} ≈ 1000 × e^{-0.2} ≈ 1000 × 0.8187 ≈ 819 nuclei.

Question 2: If a sample initially contains 500 nuclei and after 50 days only 250 nuclei remain, what is the decay constant (λ) for this isotope? Show your working.

Answer: Using N = N_0 e^{-λt}, 250 = 500 e^{-λ × 50}. Therefore, 0.5 = e^{-50λ}. Taking natural logs: ln(0.5) = -50λ. So, λ = -ln(0.5)/50 ≈ 0.01386 per day.

Question 1: Design a simple experiment to measure the decay constant of a radioactive isotope. Include the variables you would control and the safety precautions necessary.

Answer: An experiment involves measuring the activity of a known amount of radioactive material at regular intervals using a Geiger counter. Control variables include temperature, sample size, and detector calibration. Safety precautions include using shielding to reduce radiation exposure, handling samples with tongs, and wearing protective gear.

Question 2: Identify two variables that affect the rate of radioactive decay and explain how they influence the process.

Answer: Radioactive decay rate is affected by the decay constant, which is intrinsic to each isotope and does not change, and the amount of radioactive material present. The decay constant determines how quickly nuclei decay, while a larger amount results in more decay events per unit time.

Question 1: A sample of a radioactive isotope has an initial activity of 10,000 disintegrations per second. After 30 days, the activity decreases to 4,932 disintegrations per second. Calculate the decay constant (λ) based on this data.

Answer: Using activity A = A_0 e^{-λt}, 4932 = 10000 e^{-λ × 30}. So, 0.4932 = e^{-30λ}. Taking natural logs: ln(0.4932) = -30λ. Therefore, λ = -ln(0.4932)/30 ≈ 0.023 per day.

Question 2: Interpret the data: Why does the activity decrease over time, and what does this tell you about the nature of radioactive decay?

Answer: The activity decreases because the number of radioactive nuclei reduces over time due to decay. This shows that radioactive decay is a random but statistically predictable process, following an exponential decay pattern governed by the decay constant.

Question 1: Explain how decay equations are used in carbon dating to estimate the age of archaeological samples.

Answer: Decay equations relate the remaining amount of Carbon-14 in a sample to its original amount and the known half-life. By measuring the current Carbon-14 content, scientists calculate how long it has been since the sample was last alive.

Question 2: Describe a real-world industry or technology that relies on radioactive decay or decay equations, and explain its significance.

Answer: Nuclear medicine uses radioactive isotopes that decay within the body to produce images for diagnosis. Decay equations help determine safe dosage levels and timing for imaging procedures, ensuring effective and safe patient care.