Question

If the radial wavefunction of the 3p orbital of a hydrogen-like atom is

Hint:

Radial nodes are obtained by setting radial part equal to zeroIgnore exponential terms and solve polynomial part

Answer 1

Correct Answer: 6

Step 1: Condition for radial node.
Radial nodes occur where radial wavefunction becomes zero
\[ R_{31}(r) = 0 \]

Step 2: Identify zero term.
From given expression:
\[ (4 - \rho)e^{-\rho/2} = 0 \]
Since exponential term is never zero,
\[ 4 - \rho = 0 \Rightarrow \rho = 4 \]

Step 3: Use expression for $\rho$.
\[ \rho = \frac{2Z}{na_0}r \]
For $n=3$:
\[ \rho = \frac{2Z}{3a_0}r \]

Step 4: Substitute $\rho = 4$.
\[ 4 = \frac{2Z}{3a_0}r \]

Step 5: Solve for $r$.
\[ r = \frac{4 \cdot 3a_0}{2Z} = \frac{12a_0}{2Z} = \frac{6a_0}{Z} \]

Step 6: Compare with given form.
\[ r = Y \times \frac{a_0}{Z} \]
\[ Y = 6 \]

Step 7: Conclusion.
\[ \boxed{6} \]