Question

For a single electron species if its excitation energy is $Ex_{n_1}$ and separation energy is $SE_{n_2}$ then which of the following is not true? (IE = Ionisation Energy)

• A. IE = $Ex_{n_1} + SE_{n_2}(n_1 = n_2)$
• B. IE < $Ex_{n_1} + SE_{n_2}(n_1 > n_2)$
• C. IE > $Ex_{n_1} + SE_{n_2}(n_1 < n_2)$
• D. None of the above

Answer 1

Answer: (D)

None of the above

Answer 2

For a single electron species, the energy of an electron in the $n^{th}$ orbit is given by:

$E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$

Let's define the terms given in the question:

1. Ionisation Energy (IE): This is the energy required to remove an electron from its ground state ($n=1$) to an infinite distance ($n=\infty$).

   $IE = E_\infty - E_1 = 0 - (-13.6 \frac{Z^2}{1^2}) = 13.6 Z^2 \text{ eV}$

2. Excitation Energy ($Ex_{n_1}$): This is the energy required to excite an electron from its ground state ($n=1$) to an excited state ($n=n_1$).

   $Ex_{n_1} = E_{n_1} - E_1 = (-13.6 \frac{Z^2}{n_1^2}) - (-13.6 \frac{Z^2}{1^2}) = 13.6 Z^2 (1 - \frac{1}{n_1^2}) \text{ eV}$

3. Separation Energy ($SE_{n_2}$): This is the energy required to remove an electron from a specific state ($n=n_2$) to an infinite distance ($n=\infty$).

   $SE_{n_2} = E_\infty - E_{n_2} = 0 - (-13.6 \frac{Z^2}{n_2^2}) = 13.6 \frac{Z^2}{n_2^2} \text{ eV}$

Let $K = 13.6 Z^2$. Then the expressions simplify to:

$IE = K$ $Ex_{n_1} = K (1 - \frac{1}{n_1^2})$ $SE_{n_2} = \frac{K}{n_2^2}$

Now, let's evaluate the sum $Ex_{n_1} + SE_{n_2}$:

$Ex_{n_1} + SE_{n_2} = K (1 - \frac{1}{n_1^2}) + \frac{K}{n_2^2}$ $Ex_{n_1} + SE_{n_2} = K - \frac{K}{n_1^2} + \frac{K}{n_2^2}$ $Ex_{n_1} + SE_{n_2} = K + K (\frac{1}{n_2^2} - \frac{1}{n_1^2})$

Since $IE = K$, we can write:

$Ex_{n_1} + SE_{n_2} = IE + IE (\frac{1}{n_2^2} - \frac{1}{n_1^2})$

Now we check each option:

(1) IE = $Ex_{n_1} + SE_{n_2}(n_1 = n_2)$

If $n_1 = n_2$, then $\frac{1}{n_2^2} - \frac{1}{n_1^2} = 0$.

Substituting this into the derived equation:

$Ex_{n_1} + SE_{n_2} = IE + IE(0) = IE$

So, $IE = Ex_{n_1} + SE_{n_2}$ is TRUE.

(2) IE < $Ex_{n_1} + SE_{n_2}(n_1 > n_2)$

If $n_1 > n_2$, then $n_1^2 > n_2^2$.

This implies $\frac{1}{n_1^2} < \frac{1}{n_2^2}$.

Therefore, $\frac{1}{n_2^2} - \frac{1}{n_1^2} > 0$.

Since $IE = 13.6 Z^2$ is a positive value, $IE (\frac{1}{n_2^2} - \frac{1}{n_1^2})$ will be positive.

So, $Ex_{n_1} + SE_{n_2} = IE + (\text{a positive value})$.

This means $Ex_{n_1} + SE_{n_2} > IE$, or $IE < Ex_{n_1} + SE_{n_2}$.

This statement is TRUE.

(3) IE > $Ex_{n_1} + SE_{n_2}(n_1 < n_2)$

If $n_1 < n_2$, then $n_1^2 < n_2^2$.

This implies $\frac{1}{n_1^2} > \frac{1}{n_2^2}$.

Therefore, $\frac{1}{n_2^2} - \frac{1}{n_1^2} < 0$.

Since $IE = 13.6 Z^2$ is a positive value, $IE (\frac{1}{n_2^2} - \frac{1}{n_1^2})$ will be negative.

So, $Ex_{n_1} + SE_{n_2} = IE + (\text{a negative value})$.

This means $Ex_{n_1} + SE_{n_2} < IE$, or $IE > Ex_{n_1} + SE_{n_2}$.

This statement is TRUE.

Since all the statements (1), (2), and (3) are true, none of them is "not true". Therefore, the correct option is (4) "None of the above".