Problem 54 - A Complete Resource Book in Chemistry for JEE Main

Question

A radioactive form of phosphorus undergoes Y-decay. What will the radioactivity level (R) versus time graph for the decay process appear?

Step-by-Step Solution

Answer

The graph is an exponentially decreasing curve, starting at \( R_0 \) and approaching zero over time.

Step 1: Understand Y-decay

In the context of the problem, 'Y-decay' is another term for radioactive decay where the nucleus emits radiation as it transitions to a more stable state. Typically, this involves beta or gamma decay, but the exact type is not specified. However, the decay will lead to a decrease in radioactivity over time.

Step 2: Use the Radioactive Decay Formula

Radioactive decay is governed by the formula \( R(t) = R_0 e^{-kt} \), where \( R(t) \) is the radioactivity level at time \( t \), \( R_0 \) is the initial radioactivity level, \( e \) is the base of the natural logarithm, and \( k \) is the decay constant which depends on the particular isotope.

Step 3: Graph Characteristics

By examining the decay formula, the graph of radioactivity level \( R \) versus time \( t \) is a decreasing exponential function starting from \( R_0 \). It asymptotically approaches zero as \( t \) increases.

Step 4: Sketch the Graph

Draw the graph starting at \( R_0 \) on the vertical axis at time \( t = 0 \), and show a smooth curve that decreases towards zero without ever quite reaching it. The curve represents the exponential decay of the radioactive material over time.

Key Concepts

Exponential Decay

In physics, exponential decay describes the process where the quantity of a substance decreases at a rate proportional to its current value. When applied to radioactive materials, this means the material's radioactivity decreases exponentially over time. The formula for radioactive decay can be expressed as \( R(t) = R_0 e^{-kt} \). Here, \( R(t) \) represents the radioactivity at time \( t \), and \( R_0 \) is the initial amount of radioactivity. The key feature of exponential decay is its rate of decrease, which is determined by the decay constant \( k \).

• As time increases, \( e^{-kt} \) becomes very small, which is why the level of radioactivity approaches zero.
• This behavior is typical of many natural decay processes and ensures that the decline never quite reaches zero instantaneously but rather gradually diminishes to immeasurable levels.

Exponential decay can apply in various contexts such as financial depreciation, population biology, and of course, radioactive decay, making it a fundamental concept across different scientific disciplines.
Understanding each component of the exponential decay formula allows you to predict how much of the radioactive material will remain after a certain period.

Radioactivity Graph

A radioactivity graph visually represents how radioactivity decreases over time due to radioactive decay. When sketching a graph based on the formula \( R(t) = R_0 e^{-kt} \), start by plotting \( R_0 \) on the vertical axis, which represents the initial level of radioactivity when \( t = 0 \). The curve then continuously slopes downward, reflecting the material's ongoing decay.

• This downward trend is smooth and consistent because the rate of decay remains constant per unit time.
• Importantly, the graph is asymptotic, meaning that as time progresses, the curve gets closer and closer to the horizontal axis but never actually touches it.

In the context of radioactive decay, this represents the residual radioactivity eventually tending towards zero, although it theoretically never reaches complete extinguishment over finite time.
Such graphs are crucial for visualizing the decrease in radioactive substances, helping to predict the timeframe in which a certain level of radioactivity will be reached.

Decay Constant

The decay constant \( k \) is a pivotal parameter in understanding radioactive decay. It quantifies the rate at which a radioactive isotope decays, playing a key role in the formula \( R(t) = R_0 e^{-kt} \).

• The value of the decay constant is unique to each radioactive isotope because it depends on the specific properties of the substance.
• A larger decay constant means that the substance will decay more quickly, leading to a steeper decline in radioactivity over time.

The decay constant is inversely related to the half-life of the substance, which is the time taken for half of the initial material to decay. The relationship is formalized as \( k = \frac{\ln(2)}{T_{1/2}} \), where \( T_{1/2} \) represents the half-life.
This relationship helps in calculating how long it will take for a substance to decay to a certain level, providing valuable insights in various fields like nuclear physics, chemistry, and even medical treatments where radioactive isotopes are employed.