Which of the following is true ?

Options

A. According to pauli's exclasion principle ,no two electron in an atom can have the same value of quantum number n,l, and m
B. The total energy of an electron in an orbit is half of its potential energy
C. The speed of an electron in a orbit increases with increase of its quantum number n
D. The energy of an electron in a orbit decreases with increase of its quantum number n

Explanation

To determine which of the statements is true, let's analyze each statement step by step:

### Step 1: Analyze the first statement
**Statement:** According to Pauli's exclusion principle, no two electrons in an atom can have the same value of quantum number having n, l, and m.

**Analysis:**
- Pauli's exclusion principle states that no two electrons in an atom can have the same set of all four quantum numbers (n, l, m, and s).
- This statement is incorrect because it omits the spin quantum number (s).

**Conclusion:** This statement is **false**.

### Step 2: Analyze the second statement
**Statement:** The total energy of an electron in an orbit is half of its potential energy.

**Analysis:**
- The total energy (E) of an electron in an orbit is given by the relationship:
\[ E = KE + PE \]
- For a bound electron, the potential energy (PE) is negative, and it is known that:
\[ E = \frac{PE}{2} \]
- Therefore, this statement is correct.

**Conclusion:** This statement is **true**.

### Step 3: Analyze the third statement
**Statement:** The speed of an electron in an orbit increases with an increase in its quantum number n.

**Analysis:**
- The speed (v) of an electron in an orbit is given by the formula:
\[ v = \frac{Z}{n} \]
- Here, Z is the atomic number, and n is the principal quantum number. This shows that speed is inversely proportional to n.
- Therefore, as n increases, the speed decreases.

**Conclusion:** This statement is **false**.

### Step 4: Analyze the fourth statement
**Statement:** The energy of an electron in an orbit decreases with an increase in its quantum number n.

**Analysis:**
- The energy (E) of an electron in an orbit is given by:
\[ E = -\frac{13.6 Z^2}{n^2} \]
- As n increases, the value of E becomes less negative (i.e., it increases), which means the energy of the electron actually increases with increasing n.

**Conclusion:** This statement is **false**.

### Final Conclusion
The only true statement among the options provided is the second one: "The total energy of an electron in an orbit is half of its potential energy."

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