Question

A radioactive substance decays to \((1 / 16)\) th of its initial activity in 40 days. the half life of the radioactive substance expressed in day is (A) 20 (B) 5 (C) 10 (D) 4

Short answer

Given that a radioactive substance decays to \(\dfrac{1}{16}\) of its initial activity in 40 days, we first set up the radioactive decay equation: \[\dfrac{1}{16} A_{0} = A_{0}e^{-λ(40)}\] Solving for the decay constant (λ): \[λ = \dfrac{\ln \dfrac{1}{16}}{-40}\] Finally, calculating the half-life using the decay constant: \[T_{\frac{1}{2}} = \dfrac{\ln 2}{\dfrac{\ln \dfrac{1}{16}}{-40}} \] Simplifying gives us the half-life to be \(T_{\frac{1}{2}} = 10\) days. Thus, the correct answer is (C) 10.

Step-by-step solution

Given that the initial activity decays to \((1/16)\) after 40 days, the radioactive decay equation can be written as: \[ \dfrac{1}{16} A_{0} = A_{0}e^{-λ(40)}\] #Step 2: Solve the equation for the decay constant (λ)#

Divide both sides of the equation by \(A_{0}\) to get: \[ \dfrac{1}{16} = e^{-λ(40)}\] Take the natural logarithm of both sides of the equation: \[ \ln \dfrac{1}{16} = -λ(40)\] Now, we solve for \(λ\): \[ λ = \dfrac{\ln \dfrac{1}{16}}{-40} \] #Step 3: Find the half-life using the decay constant (λ)#

The half-life is given by the formula: \[ T_{\frac{1}{2}} = \dfrac{\ln 2}{λ} \] Use the decay constant found in Step 2 to calculate the half-life: \[ T_{\frac{1}{2}} = \dfrac{\ln 2}{\dfrac{\ln \dfrac{1}{16}}{-40}} \] #Step 4: Simplify and find the answer#

Simplify to get the half-life, and compare your answer with the given options: \[ T_{\frac{1}{2}} = 10\] So, the half-life of the radioactive substance expressed in days is \(10\). Therefore, the correct answer is (C) 10.

Key concepts

Radioactive Decay

Radioactive decay is a process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs in a predictable pattern, which helps scientists determine the age of fossils and rocks, among other things. The fundamental characteristic of radioactive decay is its spontaneous nature, meaning it occurs without any external influence.

When a substance is radioactive, it signifies that it contains atoms that will decay over time. It's important to note that the rate at which these atoms decay is consistent for each radioactive isotope. As the nuclei emit particles or electromagnetic waves, they transform into different elements or isotopes. This process continues until a stable nucleus is formed.

Factors affecting the radioactive decay include:

• Type of radioactive material: Different isotopes decay at different rates, measured by their half-lives.
• Environmental conditions: Though temperature and pressure generally don't affect radioactive decay, extreme conditions can influence decay rates.

The decay is represented by a mathematical equation which connects the initial quantity of a substance to the quantity present after a given time. This results in predictable and calculable decay patterns.

Exponential Decay

Exponential decay describes the phenomenon where the quantity of a substance decreases at a rate proportional to its current value. This pattern is crucial in understanding radioactive decay, where the rate of decay is proportional to the remaining amount of the substance.

Mathematically, exponential decay is expressed as:\[ N(t) = N_0 e^{-λt} \]Here, \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, and \( λ \) is the decay constant.

The nature of exponential decay means that the quantity decreases rapidly at first. Over time, the rate of decrease slows but never truly reaches zero. This can visually be seen as a curve asymptotically approaching the time axis on a graph.

Understanding exponential decay helps in fields such as:

• Radioactive dating: Estimating the age of materials.
• Medicine: Monitoring the decay of radioactive substances used in treatments.

It's crucial to grasp this concept, as it explains how substances decrease by a constant proportion over equal intervals of time.

Decay Constant

The decay constant, denoted by \( λ \) (lambda), is a parameter that indicates the speed of radioactive decay. It's a critical concept for quantifying how quickly a radioactive substance undergoes decay.

The decay constant is related to the half-life of the substance through the equation:\[ λ = \frac{\ln 2}{T_{1/2}} \]where \( T_{1/2} \) is the half-life, which signifies the time taken for half of the radioactive atoms in a sample to decay.

Higher values of \( λ \) indicate a faster rate of decay, leading to a shorter half-life. Conversely, smaller values of \( λ \) represent slower decay rates and longer half-lives.

Understanding the decay constant is essential to:

• Calculating the remaining quantity of a radioactive substance after a certain period.
• Determining the half-life from empirical decay data.

The decay constant allows for precise predictions about future quantities, which is vital for scientific applications and safety assessments in environments dealing with radioactive materials.