The graph between the instantaneous concentration (N) of a radioactive element and time (t) is.

Options

A.
B.
C.
D.

Explanation

To solve the question regarding the graph between the instantaneous concentration (N) of a radioactive element and time (t), we can follow these steps:

### Step 1: Understand the Concept of Radioactive Decay
Radioactive decay is a random process by which an unstable atomic nucleus loses energy by emitting radiation. The concentration of a radioactive substance decreases over time as it transforms into other elements or isotopes.

**Hint:** Remember that radioactive decay is a stochastic process and is characterized by a half-life.

### Step 2: Identify the Mathematical Representation
The instantaneous concentration (N) of a radioactive element at time (t) can be described by the equation:
\[ N(t) = N_0 e^{-\lambda t} \]
where:
- \( N_0 \) is the initial concentration at time \( t = 0 \),
- \( \lambda \) is the decay constant,
- \( t \) is the time.

**Hint:** The exponential function \( e^{-\lambda t} \) indicates how the concentration decreases over time.

### Step 3: Analyze the Equation
From the equation \( N(t) = N_0 e^{-\lambda t} \):
- As time (t) increases, the term \( e^{-\lambda t} \) decreases.
- This means that the instantaneous concentration \( N \) decreases exponentially.

**Hint:** Focus on the negative exponent; it signifies a decrease in concentration over time.

### Step 4: Graphical Representation
When we plot the graph of \( N \) versus \( t \):
- The y-axis represents the instantaneous concentration \( N \).
- The x-axis represents time \( t \).
- The curve will start at \( N_0 \) when \( t = 0 \) and will approach zero as \( t \) increases, forming a smooth, downward-sloping curve.

**Hint:** The shape of the graph is crucial; it should show a rapid decrease initially that gradually slows down.

### Step 5: Conclusion
Based on the analysis, the graph of the instantaneous concentration \( N \) of a radioactive element versus time \( t \) is an exponential decay curve. Therefore, the correct option is that the graph shows an exponential decrease in concentration over time.

**Final Answer:** The graph between the instantaneous concentration (N) of a radioactive element and time (t) is an exponential decay curve.