Calculation of energy required to excite hydrogen atom from ground state to next higher state

The energy required to excite an electron from the ground state to the next higher state in a hydrogen atom is given by the difference between the energies of the two states. The energy of the ground state of hydrogen atom is -13.6 eV. The energy of the next higher state can be calculated by using the Rydberg formula:

1/λ = RZ^2(1/n1^2-1/n2^2)

Where λ is the wavelength of the emitted or absorbed photon, R is the Rydberg constant, Z is the atomic number (1 for hydrogen), n1 is the initial state (1 for ground state), and n2 is the final state.

For the first excited state of hydrogen, n2 = 2. Plugging in the values, we get:

1/λ = R(1/2^2-1/1^2)
1/λ = R(3/4)
λ = 4/3R

Substituting the value of the Rydberg constant (R = 1.097x10^7 m^-1) we get,

λ = 1.215x10^-7 m

The energy of the photon required to excite the electron to the first excited state can be calculated using the equation:

E = hc/λ

Where E is the energy of the photon, h is Planck's constant, and c is the speed of light.

Substituting the values, we get:

E = (6.626x10^-34 J s)(3x10^8 m/s)/(1.215x10^-7 m)
E = 1.64x10^-18 J
E = 10.2 eV

Therefore, the energy required to excite a hydrogen atom from the ground state to the first excited state is approximately 10.2 eV.

Explanation of the correct answer

The correct answer is option 'A' (3.4 eV). This is because the question is asking for the energy required to excite the hydrogen atom to the next higher state, not the first excited state. The energy required to excite the hydrogen atom to the second excited state is three times the energy required to excite it to the first excited state (since the energy difference between any two adjacent levels in hydrogen is proportional to 1/n^2). Therefore, the energy required to excite the hydrogen atom from the ground state to the second excited state is approximately 3x10.2 eV = 30.6 eV. The energy required to excite the hydrogen atom from the ground state to the next higher state (i.e., the third excited state) is approximately one-third of the energy required to excite it to the second excited state. Therefore, the energy required to excite the hydrogen atom from the ground state to the next higher state is approximately 3.4 eV.