Which of the following is false?

Options

A. The orbitals are no more degenerate in the presence of a magnetic field
B. The spin quantum number was introduced to explain the splitting of spectral lines of hydrogen atom in the presence of a magnetic field
C. Pauil exclusion principle is followed by fermions which have half integral spins
D. The energy of an orbitals in an atom remains the same with increases in the positive charge in its nucleus

Explanation

To determine which of the given statements is false, let's analyze each statement one by one:

### Step 1: Analyze the First Statement
**Statement:** The orbitals are no more degenerate in the presence of a magnetic field.

**Analysis:** This statement is true. In the presence of an external magnetic field, the energy levels of orbitals split due to the Zeeman effect, meaning they are no longer degenerate (i.e., they do not have the same energy).

### Step 2: Analyze the Second Statement
**Statement:** The spin quantum number was introduced to explain the splitting of spectral lines of the hydrogen atom in the presence of a magnetic field.

**Analysis:** This statement is also true. The introduction of the spin quantum number (which can take values of +1/2 or -1/2) helps explain the splitting of spectral lines (known as fine structure) in the presence of a magnetic field.

### Step 3: Analyze the Third Statement
**Statement:** Pauli's exclusion principle is followed by fermions which have half-integral spins.

**Analysis:** This statement is true. Fermions, which include particles like electrons, follow Pauli's exclusion principle, which states that no two fermions can occupy the same quantum state simultaneously.

### Step 4: Analyze the Fourth Statement
**Statement:** The energy of an orbital in an atom remains the same with an increase in positive charge in its nucleus.

**Analysis:** This statement is false. The energy of an orbital is affected by the nuclear charge (Z). According to the formula for the energy of an electron in a hydrogen-like atom, as the nuclear charge increases, the energy of the orbital becomes more negative (i.e., the energy decreases). Therefore, this statement is incorrect.

### Conclusion
The false statement is the fourth one: "The energy of an orbital in an atom remains the same with an increase in positive charge in its nucleus."

### Final Answer
The correct option is **D**.

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