Calculation of the most probable radius of an electron in a 1s orbital of a Ne atom (Ne9):

To calculate the most probable radius at which an electron will be found when it occupies a 1s orbital of a Ne atom (Ne9), we need to consider the concept of radial probability distribution. The radial probability distribution gives the probability of finding an electron at a particular distance from the nucleus.

The equation for the radial probability distribution of the 1s orbital is given by:

P(r) = 4πr²R(r)²

Where:
- P(r) represents the radial probability distribution
- r represents the distance from the nucleus
- R(r) represents the radial wave function

The radial wave function for the 1s orbital is given by:

R(r) = (2/Z³)^(1/2) * e^(-r/Z)

Where:
- Z represents the atomic number of the element (Ne in this case)
- e is the base of natural logarithm (~2.71828)

The most probable radius is the value of r at which the radial probability distribution, P(r), is maximum. In other words, we need to find the maximum value of P(r) by differentiating it with respect to r and equating it to zero.

Key Points:
- The most probable radius corresponds to the maximum of the radial probability distribution.
- The radial probability distribution is given by P(r) = 4πr²R(r)².
- The radial wave function for the 1s orbital is given by R(r) = (2/Z³)^(1/2) * e^(-r/Z).
- To find the most probable radius, differentiate P(r) with respect to r and equate it to zero.
- Solve the resulting equation to obtain the most probable radius.

Calculation:

To find the maximum of the radial probability distribution, we need to differentiate P(r) with respect to r and equate it to zero.

d(P(r))/dr = 0

Differentiating P(r) = 4πr²R(r)² with respect to r gives:

d(P(r))/dr = 8πrR(r)² + 8πr²R(r)R'(r) = 0

Where R'(r) represents the derivative of the radial wave function with respect to r.

Simplifying the equation:

8πrR(r)² + 8πr²R(r)R'(r) = 0

rR(r) + r²R'(r) = 0

Solving the resulting equation will give us the most probable radius.

Key Points:
- Differentiate P(r) with respect to r and equate it to zero.
- Simplify the resulting equation.
- Solve the equation to obtain the most probable radius.

Conclusion:

By solving the equation rR(r) + r²R'(r) = 0, we can find the most probable radius at which an electron will be found when it occupies a 1s orbital of a Ne atom (Ne9). The correct answer is between 5.27 pm and 5.31 pm.