What is the distance of separation between 3rd and 4th orbit of H-atom ?

Answer

To find the distance of separation between the 3rd and 4th orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the formula for the radius of the hydrogen atom's orbit.
The radius of the nth orbit of a hydrogen atom is given by the formula:
\[ r_n = 0.529 \, \text{Å} \times n^2 \]
where \( r_n \) is the radius of the nth orbit and \( n \) is the principal quantum number (the orbit number).

### Step 2: Calculate the radius of the 4th orbit.
Using the formula for the 4th orbit (\( n = 4 \)):
\[ r_4 = 0.529 \, \text{Å} \times 4^2 \]
\[ r_4 = 0.529 \, \text{Å} \times 16 \]
\[ r_4 = 8.464 \, \text{Å} \]

### Step 3: Calculate the radius of the 3rd orbit.
Using the formula for the 3rd orbit (\( n = 3 \)):
\[ r_3 = 0.529 \, \text{Å} \times 3^2 \]
\[ r_3 = 0.529 \, \text{Å} \times 9 \]
\[ r_3 = 4.761 \, \text{Å} \]

### Step 4: Find the distance of separation between the 3rd and 4th orbits.
To find the distance between the 3rd and 4th orbits, we subtract the radius of the 3rd orbit from the radius of the 4th orbit:
\[ \text{Distance} = r_4 - r_3 \]
\[ \text{Distance} = 8.464 \, \text{Å} - 4.761 \, \text{Å} \]
\[ \text{Distance} = 3.703 \, \text{Å} \]

### Step 5: Convert the distance to picometers.
1 Ångström (Å) is equal to \( 10^{-10} \) meters, and 1 picometer (pm) is equal to \( 10^{-12} \) meters. Therefore, to convert Ångströms to picometers:
\[ 3.703 \, \text{Å} = 3.703 \times 10^{-10} \, \text{m} \]
To convert to picometers:
\[ 3.703 \times 10^{-10} \, \text{m} = 3.703 \times 10^{-10} \times 10^{2} \, \text{pm} \]
\[ = 3.703 \, \text{pm} \]

### Final Answer:
The distance of separation between the 3rd and 4th orbit of the hydrogen atom is approximately **3.703 Å** or **3.703 pm**.

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