The radial distribution function `[P(r)]` is used to determine the most probble radius, which is used to find the electron in a givenorbital. `(dp(r))/(dr)` for `1s`- orbital of hydrogen like atom having atomic number Z, is `(dp)/(dr)=(4Z^(3))/(a_(o)"^(3))(2r-(2Zr^(2))/(a_(o)))e^(-2zr//a_(o)):`
Then which of the following statements is/are correct ?

Options

A. At the point of maximum value of radial distribution function `(dp(r))/(dr)=0,` one antionde is present
B. Most probable radius of `Li^(2+)isa_(o)/(3)"pm"`
C. Most probable radus of `He^(+)isa_(o)/(2)"pm"`
D. Most probable radius of hydrogen atom is `a_(o)` pm

Answer

To solve the problem regarding the radial distribution function \( P(r) \) for the 1s orbital of a hydrogen-like atom, we need to analyze the given expression for \( \frac{dP(r)}{dr} \) and determine the most probable radius. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Radial Distribution Function
The radial distribution function \( P(r) \) describes the probability of finding an electron at a distance \( r \) from the nucleus in a given orbital. For the 1s orbital of a hydrogen-like atom, the expression for \( \frac{dP(r)}{dr} \) is given as:

\[ \frac{dP(r)}{dr} = \frac{4Z^3}{a_0^3} \left( 2r - \frac{2Zr^2}{a_0} \right) e^{-\frac{2Zr}{a_0}} \]

### Step 2: Set the Derivative to Zero
To find the most probable radius, we need to find the value of \( r \) where \( \frac{dP(r)}{dr} = 0 \). This occurs at the maximum value of the radial distribution function.

Setting the equation to zero:

\[ \frac{4Z^3}{a_0^3} \left( 2r - \frac{2Zr^2}{a_0} \right) e^{-\frac{2Zr}{a_0}} = 0 \]

Since \( e^{-\frac{2Zr}{a_0}} \) is never zero, we focus on the term inside the parentheses:

\[ 2r - \frac{2Zr^2}{a_0} = 0 \]

### Step 3: Solve for \( r \)
Rearranging the equation:

\[ 2r = \frac{2Zr^2}{a_0} \]

Dividing both sides by \( 2r \) (assuming \( r \neq 0 \)):

\[ 1 = \frac{Zr}{a_0} \]

Thus, we find:

\[ r = \frac{a_0}{Z} \]

This gives us the most probable radius for the electron in the 1s orbital of a hydrogen-like atom.

### Step 4: Evaluate the Statements
Now we evaluate the correctness of the statements provided:

1. **Statement 1**: At the point of maximum value of radial distribution function \( \frac{dP(r)}{dr} = 0 \) is correct.
2. **Statement 2**: The most probable radius of \( \text{Li}^{2+} \) (Z=3) is \( \frac{a_0}{3} \) picometers, which is correct.
3. **Statement 3**: The most probable radius of \( \text{He}^+ \) (Z=2) is \( \frac{a_0}{2} \) picometers, which is also correct.
4. **Statement 4**: The most probable radius of hydrogen (Z=1) is \( a_0 \) picometers, which is correct.

### Conclusion
All statements are correct. Therefore, the answer is that all statements (A, B, C, D) are correct.

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