Based on equation `E = -2.178 xx 10^-18 J((Z^2)/(n^2))`, certain conclusions are written. Which of them is not correct ?

Options

A. Larger the value of n , the larger is the orbit radius
B. Equation can be used to Calculate the change in energy when the electron changes orbit
C. For n=1 , the electron has a more negative energy than it does for n=6 which means that the electron is more loosely bound in the smallest allowed orbit
D. The negative sign in equation simpty means that the energy of electron bound to the nucleus is lower than it would be if the electrons were at the infinite distance from the nucleus

Answer

To determine which conclusion based on the equation \( E = -2.178 \times 10^{-18} J \left( \frac{Z^2}{n^2} \right) \) is incorrect, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement 1:** "Larger the value of n, larger is the orbit radius."

- The principal quantum number \( n \) indicates the energy level of an electron in an atom. As \( n \) increases, the radius of the orbit also increases.
- **Conclusion:** This statement is **correct**.

### Step 2: Analyze the second statement
**Statement 2:** "Equation can be used to calculate the change in energy when the electron changes orbit."

- The energy \( E \) given by the equation depends on \( n \), the principal quantum number. When an electron transitions between orbits (i.e., changes \( n \)), the energy changes accordingly.
- **Conclusion:** This statement is **correct**.

### Step 3: Analyze the third statement
**Statement 3:** "For \( n = 1 \), the electron has more negative energy than for \( n = 6 \), which means the electron is loosely bound in the smallest allowed orbital."

- For \( n = 1 \):
\[ E_1 = -2.178 \times 10^{-18} \times \frac{Z^2}{1^2} = -2.178 \times 10^{-18} J \cdot Z^2 \]

- For \( n = 6 \):
\[ E_6 = -2.178 \times 10^{-18} \times \frac{Z^2}{6^2} = -2.178 \times 10^{-18} \times \frac{Z^2}{36} \]

- Since \( E_1 \) is more negative than \( E_6 \), it indicates that the electron in the \( n = 1 \) state is more tightly bound than in the \( n = 6 \) state.
- **Conclusion:** The statement is **incorrect** because it claims the electron is loosely bound in the smallest allowed orbital.

### Step 4: Analyze the fourth statement
**Statement 4:** "The negative sign in the equation simply means that the energy of an electron bound to the nucleus is lower than it would be if the electron were at the infinite distance from the nucleus."

- The negative energy indicates that the electron is in a bound state. At infinite distance, the energy is zero, and being bound means the energy is lower (more negative).
- **Conclusion:** This statement is **correct**.

### Final Conclusion
The incorrect statement among the provided options is **Statement 3**.

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