# The expression for the energy of the electron in an orbit is given by $E = - \frac{e^{2}}{8\pi\epsilon_{0}r}$, then which of the following statement

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The expression for the energy of the electron in an orbit is given by $E = - \frac{e^{2}}{8\pi\epsilon_{0}r}$, then which of the following statement is/are true regarding the equation

(i)  The negative sign in the equation implies electron is trapped in the atom

(ii) If energy equal to $E = +\frac{e^{2}}{8\pi\epsilon_{0}r}$is given the electrons can be released from the atom

(iii) If the electron is away from the nucleus then the energy increases

(iv) The total energy of the electron depend only on the orbital radius

1. Only (i) & (ii)
2. Only (ii) & (iii)
3. All the above
4. Only (i) and (iv)

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Correct Answer - Option 3 : All the above

CONCEPT

• The kinetic energy of the electron in an orbit is given by

$⇒ KE = \frac{1}{2}mV^{2} = \frac{1}{2} m \times \frac{e ^{2}}{4\pi \epsilon_{0}mr} =\frac{e^{2}}{8\pi\epsilon_{0}mr}$

• The potential energy of an electron in an orbit is given by

$⇒ U = -\frac{e^{2}}{4\pi\epsilon_{0}r}$

• The total energy of the electron is given by

⇒ T = KE + U

$⇒ T = \frac{e^{2}}{8\pi\epsilon_{0}r} - \frac{e^{2}}{4\pi\epsilon_{0}r} = -\frac{e^{2}}{8\pi \epsilon_{0}r}$

Where e= charge, r = orbital radius, T = Total energy

EXPLANATION:

• The total energy of an electron in an atom is given by

$⇒ T = -\frac{e^{2}}{8\pi \epsilon_{0}r}$

• By analyzing the above equation it is clear that the total energy is inversely proportional to the distance, which in turn means that
• As the value of r changes the value of T changes, which justifies statement 4
•  As the value of r increases the value of T increases, which justifies statement 3.    [ Kindly note the negative sign in the equation of T, $T = -\frac{e^{2}}{8\pi \epsilon_{0}r}$
• By giving an amount of energy that equal to the energy of an electron in an atom the electron can be released from it.
• Since the energy of an electron in an atom is $T = -\frac{e^{2}}{8\pi \epsilon_{0}r}$if we gave energy $E = +\frac{e^{2}}{8\pi\epsilon_{0}r}$then the electron can be released from the atom. Hence, statement 2  is correct
• The energy of an electron in an atom is   $T = -\frac{e^{2}}{8\pi \epsilon_{0}r}$ , by analyzing the equation means the electron is having negative energy, which in turn means the electron can't escape from the atom. Hence, statement 1 is correct
• The option which states all the statement is correct is the answer, Hence, option 3 is the answer
• Options 1, 2, and 3 doesn't contain all the statements. Hence these options are incorrect