Problem 33 - Master Resource Book in JEE Main Physics

Question

A sample of an element is \(10.38 \mathrm{~g}\). If half-life of element is \(3.8\) days, then after 19 days, how much quantity of element remains? (a) \(0.151 \mathrm{~g}\) (b) \(0.32 \mathrm{~g}\) (c) \(1.51 \mathrm{q}\) (d) \(0.16 \mathrm{q}\)

Step-by-Step Solution

Answer

After 19 days, 0.32 g of the element remains. The correct answer is (b).

Step 1: Understand the half-life formula

The half-life formula is used to calculate the remaining quantity of a substance after a given time period. The formula is: \[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{t/T} \] where \(N(t)\) is the remaining quantity, \(N_0\) is the initial quantity (in this case, 10.38 g), \(t\) is the time elapsed (19 days), and \(T\) is the half-life of the substance (3.8 days).

Step 2: Insert the given values into the formula

Replace the values in the formula with the given numbers: \[ N(19) = 10.38 \times \left(\frac{1}{2}\right)^{19/3.8} \] Here, we have substituted \(N_0 = 10.38\) grams, \(t = 19\) days, and \(T = 3.8\) days.

Step 3: Calculate the exponent

First, calculate the value of the exponent \(\frac{19}{3.8}\). \[ \frac{19}{3.8} = 5 \] So the exponent becomes 5, indicating that approximately five half-lives have passed.

Step 4: Calculate quantity remaining

Now calculate the remaining quantity of the element: \[ N(19) = 10.38 \times \left(\frac{1}{2}\right)^{5} \] Since \(\left(\frac{1}{2}\right)^{5} = \frac{1}{32}\), we multiply 10.38 by this fraction: \[ N(19) = 10.38 \times \frac{1}{32} \approx 0.324375 \] The remaining quantity of the element is approximately 0.32 g.

Step 5: Round and match to options

Round the calculated value to match the closest option given in the problem: The value 0.324375 can be rounded to 0.32 g, which matches option (b).

Key Concepts

Radioactive Decay: A Natural Transformation

Radioactive decay is a process by which an unstable atomic nucleus loses energy by emitting radiation. This can involve the emission of alpha particles, beta particles, or gamma rays. As these elements decay, they transmogrify into different elements or isotopes, altering their atomic structure.

Understanding radioactive decay is essential in fields such as nuclear physics, archaeology, and medicine. Here's how it breaks down:

• Unstable Nuclei: Certain isotopes of elements have unstable nuclei due to an imbalance in the forces holding the nucleus together.
• Decay Into Stability: Over time, these unstable nuclei will decay into a more stable form, often transforming into a different element or isotope.
• Decay Series: Once decay starts, it can form a chain of transformations, known as a decay series.

Through radioactive decay, scientists can work out the ages of rocks or fossils and analyze the composition of ancient artifacts. Each element has its unique decay rate, often defined by a parameter known as the half-life, which plays a vital role in these calculations.

Unraveling Exponential Decay

Exponential decay is a process where the rate of decay of a quantity is proportional to its current value. Simply put, the larger the quantity, the faster it decreases over time.

In the case of radioactive substances, the amount remaining reduces by half over a constant time period known as the half-life. This results in a curve where the decrease happens rapidly at first and then slows over time:

• The Half-life Concept: The half-life is the time it takes for half of the substance to decay.
• Mathematical Representation: Exponential decay is represented by the formula:i[t] = i_0 \times \left(\frac{1}{2}\right)^{t/T}where \(i[t]\) is the remaining substance at time \(t\), \(i_0\) is the initial amount, and \(T\) is the half-life.
• Understanding the Rate: This means that after one half-life, only 50% of the original amount remains. After two half-lives, 25% remains, and so on.

For the problem at hand, it took five half-lives to reduce 10.38 grams of an element down to about 0.32 grams, illustrating how exponential decay functions over multiple periods.

Nuclear Physics: Exploring the Core

Nuclear physics is the branch of physics that deals with the constituents and interactions of atomic nuclei. It dives into understanding the very heart of atoms and how they behave. One of the key phenomena studied in this field is radioactive decay.

Nuclear physics outlines the principles governing *why* and *how* nuclei decay:

• Forces at Play: Inside an atomic nucleus, powerful forces bind protons and neutrons together, known as the strong nuclear force.
• Instability Leading to Decay: If the forces in a nucleus become imbalanced, the nucleus may become unstable and undergo decay to achieve stability.
• Advancements and Applications: Insights from studying nuclear physics have led to applications in nuclear power, radiation therapies in medicine, and more.

By learning about the nuclei and their behaviors, scientists harness the power to understand natural processes on both minute and grand scales such as the functioning of the sun or the age of ancient materials.