The state `S_(1)` is

Options

A. 1s
B. 2s
C. 2p
D. 3s

Explanation

To determine the state \( S_1 \), we can follow these steps:

### Step 1: Understand the notation
The notation \( S_1 \) indicates that we are dealing with an spherically symmetrical state. In atomic orbitals, the "s" refers to the s orbital, which is known for its spherical shape.

**Hint:** Remember that "s" orbitals are spherical in shape.

### Step 2: Identify the possible options
The options given are:
1. 1S
2. 2S
3. 2P
4. 3S

Since \( S_1 \) refers to an s orbital, we can eliminate option 3 (2P) because P orbitals are not spherical.

**Hint:** Focus on the options that contain "S" as they represent s orbitals.

### Step 3: Determine the number of nodes
The \( S_1 \) state is said to have one radial node. The number of radial nodes in an orbital can be calculated using the formula:
\[ \text{Number of radial nodes} = n - l - 1 \]
where \( n \) is the principal quantum number and \( l \) is the azimuthal quantum number. For s orbitals, \( l = 0 \).

**Hint:** Recall that for s orbitals, \( l \) is always 0.

### Step 4: Apply the formula
For the \( S_1 \) state:
\[ \text{Number of radial nodes} = n - 0 - 1 = n - 1 \]
Since it is given that there is one radial node, we can set up the equation:
\[ n - 1 = 1 \]
Solving this gives:
\[ n = 2 \]

**Hint:** Solve for \( n \) to find the principal quantum number.

### Step 5: Identify the correct state
Since we found \( n = 2 \), this corresponds to the 2s orbital. Thus, the state \( S_1 \) is 2S.

**Hint:** Match the value of \( n \) to the appropriate orbital designation.

### Final Answer
The state \( S_1 \) is **2S**.