Question

Which of the following electron transitions in a hydrogen atom will require the largest amount of energy?
A)from n = 1 to n = 2
B)from n = 2 to n = 3
C)from n = $\mathrm{\infty }$ to n = 1
D)from n = 3 to n = 5

Answer 1

Hint: Energy is given out by atoms when an electron jumps from one shell to another. The amount of energy required by the atom is more if the electron jumps from the shell which is closer to the nucleus. Also, the energy required is more if the quantum numbers are alternate numbers rather than being consecutive numbers.

Complete Step By Step Answer:
We know that electrons are present in the atom in the orbits around the nucleus. These orbits are also called shells. The electrons revolve around the nucleus because of the nuclear force of attraction.
In Option C the electron moves from shell number $\mathrm{\infty }$ to 1 and in Option D, the electron moves from shell number 3 to 1. From the above-given options, we will eliminate option 3 and option 4 because the shells are not close to the nucleus. Also, the closer the shell is to the nucleus less is the possessed potential energy by the atom.
To find the amount of energy released we will use the formula,
 $\mathrm{\Delta }\text{ E = }\left[{\displaystyle \frac{1}{{(n_1)}^2}}\text{ - }{\displaystyle \frac{1}{{(n_2)}^2}}\right]$
Where n is the principal quantum number.
Now let's see the energy required by the atom when the electron transitions from n = 1 to n = 2.
 $\mathrm{\Delta }\text{ E = }\left[{\displaystyle \frac{1}{{(1)}^2}}\text{ - }{\displaystyle \frac{1}{{(2)}^2}}\right]$
 $=\left[{\displaystyle \frac{1}{1}}\text{ - }{\displaystyle \frac{1}{4}}\right]$
 $=\left[{\displaystyle \frac{3}{4}}\right]$
 $=\text{ 0}\text{.75}$
Now, let's see the energy required by the atom when the electron transitions from n = 2 to n = 3.
 $\mathrm{\Delta }\text{ E = }\left[{\displaystyle \frac{1}{{(2)}^2}}\text{ - }{\displaystyle \frac{1}{{(3)}^2}}\right]$
 $=\left[{\displaystyle \frac{1}{4}}\text{ - }{\displaystyle \frac{1}{9}}\right]$
 $=\left[{\displaystyle \frac{5}{36}}\right]$
 $=\text{ 0}\text{.138}$
 $\approx \text{ 0}\text{.14}$
The amount of energy required by the atom by the transition from n = 1 to n = 2 is 0.75 and the amount of energy required by the atom by the transition from n = 2 to n = 3 is 0.14.
The electron transition from n=1 to n=2 in an atom will require the largest amount of energy.
Final answer: The correct answer is Option A= from n = 1 to n = 2.

Note:
Remember that if the electron transitions occur from a higher energy level to a lower energy level, energy is emitted by the atom in the form of photons. However, if the electron transitions occur from a lower energy level to a higher energy level the energy is required by the atoms.