Hydrogen-like Ion Atomic Number Calculation
The energy of an electron in the n-th state of a hydrogen-like ion with atomic number Z is given by the formula:
$E_n = -13.6 \times \frac{Z^2}{n^2}$ eV
Determine Quantum Numbers
- The ground state corresponds to the principal quantum number, $n = 1$.
- The first excitation state corresponds to $n = 2$.
- The second excitation state corresponds to $n = 3$.
Calculate Energy Difference
The energy difference is given between the 2nd excitation state ($n=3$) and the ground state ($n=1$). We calculate the energy required to transition from the ground state to the second excited state:
- Energy of the ground state ($n=1$): $E_1 = -13.6 \times \frac{Z^2}{1^2}$ eV
- Energy of the 2nd excitation state ($n=3$): $E_3 = -13.6 \times \frac{Z^2}{3^2} = -13.6 \times \frac{Z^2}{9}$ eV
- The energy difference ($\Delta E$) is the difference between the higher energy state and the lower energy state:
$\Delta E = E_3 - E_1 = \left(-13.6 \times \frac{Z^2}{9}\right) - \left(-13.6 \times Z^2\right)$
$\Delta E = -13.6 \times Z^2 \times \left(\frac{1}{9} - 1\right) = -13.6 \times Z^2 \times \left(-\frac{8}{9}\right)$
$\Delta E = 13.6 \times Z^2 \times \frac{8}{9}$ eV
Solve for Atomic Number (Z)
We are given that the energy difference is 108.8 eV.
Set the calculated energy difference equal to the given value:
$13.6 \times Z^2 \times \frac{8}{9} = 108.8$
Rearrange the equation to solve for $Z^2$:
$Z^2 = \frac{108.8 \times 9}{13.6 \times 8}$
Calculate the value:
$13.6 \times 8 = 108.8$
$Z^2 = \frac{108.8 \times 9}{108.8}$
$Z^2 = 9$
Take the square root to find Z:
$Z = \sqrt{9} = 3$
Conclusion
The atomic number of the hydrogen-like ion is 3.
Similar Questions
Match the LIST-I with LIST-II
LIST-I LIST-II A. $^1_0n + ^{235}_{92}U \rightarrow ^{140}_{54}Xe + ^{94}_{38}Sr + 2^1_0n$ I. Chemical reaction B. $2H_2+O_2\rightarrow 2H_2O$ II. Fusion with +ve Q value C. $^1_1H+^2_1H \rightarrow ^3_2He + ^1_0n$ III. Fission D. $^1_1H+^1_1H \rightarrow ^2_1H+^0_1e$ IV. Fusion with -ve Q value Choose the correct answer from the options given below:
Choose the correct logic circuit for the given truth table having inputs A and B.
\[\begin{array}{|c|c|c|} \hline A & B & Y \\ \hline 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \hline \end{array}\]Two coherent monochromatic light beams of intensities 4I and 9I are superimposed. The difference between the maximum and minimum intensities in the resulting interference pattern is xI. The value of x is __________
Considering Bohr's atomic model for hydrogen atom :
(A) the energy of H atom in ground state is same as energy of $He^+$ ion in its first excited state.
(B) the energy of H atom in ground state is same as that for $Li^{++}$ ion in its second excited state.
(C) the energy of H atom in ground state is same as that of $He^+$ ion for its ground state.
(D) the energy of $He^+$ ion in its first excited state is same as that for $Li^{++}$ ion in its ground state.
Choose the correct answer from the options given below :A monochromatic light is incident on a metallic plate having work function $\phi$. An electron, emitted normally to the plate from a point A with maximum kinetic energy, enters a constant magnetic field, perpendicular to the initial velocity of electron. The electron passes through a curve and hits back the plate at a point B. The distance between A and B is:
(Given: The magnitude of charge of an electron is $e$ and mass is $m$, $h$ is Planck's constant and $c$ is velocity of light. Take the magnetic field exists throughout the path of electron)- A zener diode with 5 V zener voltage is used to regulate an unregulated dc voltage input of 25 V. For a 400 $\Omega$ resistor connected in series, the zener current is found to be 4 times load current. The load current ($I_L$) and load resistance ($R_L$) are :
- Given below are two statements :
Statement (I) : The dimensions of Planck's constant and angular momentum are same.
Statement (II): In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant.
In the light of the above statements, choose the most appropriate answer from the options given below : - A radioactive material P first decays into Q and then Q decays to non-radioactive material R. Which of the following figure represents time dependent mass of P, Q and R ?
- Consider a n-type semiconductor in which $n_e$ and $n_h$ are number of electrons and holes, respectively.
(A) Holes are minority carriers
(B) The dopant is a pentavalent atom
(C) $n_e n_h \neq n_i^2$
(where $n_i$ is number of electrons or holes in semiconductor when it is intrinsic form)
(D) $n_e n_h = n_i^2$
(E) The holes are not generated due to the donorsChoose the correct answer from the options given below : - For a hydrogen atom, the ratio of the largest wavelength of Lyman series to that of the Balmer series is
Important Questions from Modern Physics
Match the LIST-I with LIST-II
LIST-I LIST-II A. $^1_0n + ^{235}_{92}U \rightarrow ^{140}_{54}Xe + ^{94}_{38}Sr + 2^1_0n$ I. Chemical reaction B. $2H_2+O_2\rightarrow 2H_2O$ II. Fusion with +ve Q value C. $^1_1H+^2_1H \rightarrow ^3_2He + ^1_0n$ III. Fission D. $^1_1H+^1_1H \rightarrow ^2_1H+^0_1e$ IV. Fusion with -ve Q value Choose the correct answer from the options given below:
Choose the correct logic circuit for the given truth table having inputs A and B.
\[\begin{array}{|c|c|c|} \hline A & B & Y \\ \hline 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \hline \end{array}\]Two coherent monochromatic light beams of intensities 4I and 9I are superimposed. The difference between the maximum and minimum intensities in the resulting interference pattern is xI. The value of x is __________
Considering Bohr's atomic model for hydrogen atom :
(A) the energy of H atom in ground state is same as energy of $He^+$ ion in its first excited state.
(B) the energy of H atom in ground state is same as that for $Li^{++}$ ion in its second excited state.
(C) the energy of H atom in ground state is same as that of $He^+$ ion for its ground state.
(D) the energy of $He^+$ ion in its first excited state is same as that for $Li^{++}$ ion in its ground state.
Choose the correct answer from the options given below :A monochromatic light is incident on a metallic plate having work function $\phi$. An electron, emitted normally to the plate from a point A with maximum kinetic energy, enters a constant magnetic field, perpendicular to the initial velocity of electron. The electron passes through a curve and hits back the plate at a point B. The distance between A and B is:
(Given: The magnitude of charge of an electron is $e$ and mass is $m$, $h$ is Planck's constant and $c$ is velocity of light. Take the magnetic field exists throughout the path of electron)
