The plot of radial probability density ($4\pi r^2 R^2$) against $r$ for an electron in $np$ orbital of a many electron atom is given below. The value of $n$ is
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On a radial probability distribution plot ($4\pi r^2 R^2$ vs $r$):
- The number of peaks is equal to $n - l$.
- The number of radial nodes (troughs touching the baseline, excluding the origin) is $n - l - 1$.
For this plot, there are 2 peaks, so $n - l = 2$. For a $p$ orbital ($l = 1$), this immediately gives $n = 3$.
Step 1: Understanding the Question:
We need to determine the principal quantum number $n$ for an $np$ orbital from the given radial probability density curve.
Step 2: Key Formula or Approach:
The number of radial nodes for any given orbital is given by the formula:
\[ \text{Radial Nodes} = n - l - 1 \]
Where:
- $n$ is the principal quantum number.
- $l$ is the azimuthal quantum number. For a $p$ orbital, $l = 1$.
Step 3: Detailed Explanation:
From the given plot of radial probability density ($4\pi r^2 R^2$) against $r$, we count the number of times the curve touches the zero line (excluding $r = 0$ and $r \rightarrow \infty$).
The graph shows exactly one point between the two peaks where the probability density drops to zero.
This means the number of radial nodes is equal to $1$.
Applying the formula for radial nodes:
\[ \text{Radial Nodes} = n - l - 1 = 1 \]
For a $p$ orbital, $l = 1$:
\[ n - 1 - 1 = 1 \]
\[ n - 2 = 1 \]
\[ n = 3 \]
Thus, the orbital is $3p$, and the value of $n$ is 3.
