Hydrogen Atom Spectral Series Wavelength Calculation
This solution calculates the ratio of the largest wavelengths for the Lyman and Balmer series in the hydrogen atom using the Rydberg formula.
Rydberg Formula Application
The wavelength ($\lambda$) of a spectral line emitted by a hydrogen atom is given by the Rydberg formula:
$ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) $Where '$R$' is the Rydberg constant, '$n_f$' is the principal quantum number of the final energy level, and '$n_i$' is the principal quantum number of the initial energy level. The largest wavelength ($\lambda_{max}$) corresponds to the smallest energy difference, which occurs for the transition from the lowest possible initial state to the final state.
Lyman Series Maximum Wavelength
The Lyman series involves transitions to the final state $n_f = 1$. The largest wavelength occurs when the electron transitions from the next lowest state, $n_i = 2$.
Using the formula:
$ \frac{1}{\lambda_{L,max}} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) $Therefore, the largest wavelength for the Lyman series is:
$ \lambda_{L,max} = \frac{4}{3R} $Balmer Series Maximum Wavelength
The Balmer series involves transitions to the final state $n_f = 2$. The largest wavelength occurs when the electron transitions from the next lowest state, $n_i = 3$.
Using the formula:
$ \frac{1}{\lambda_{B,max}} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{9 - 4}{36} \right) = R \left( \frac{5}{36} \right) $Therefore, the largest wavelength for the Balmer series is:
$ \lambda_{B,max} = \frac{36}{5R} $Ratio Calculation
The question asks for the ratio of the largest wavelength of the Lyman series to that of the Balmer series:
$ \frac{\lambda_{L,max}}{\lambda_{B,max}} = \frac{\left( \frac{4}{3R} \right)}{\left( \frac{36}{5R} \right)} $Simplify the ratio:
$ \frac{\lambda_{L,max}}{\lambda_{B,max}} = \frac{4}{3R} \times \frac{5R}{36} = \frac{4 \times 5}{3 \times 36} = \frac{20}{108} $Reducing the fraction gives:
$ \frac{20}{108} = \frac{5}{27} $The ratio is $5:27$.
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 ?
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(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) $n_e n_h = n_i^2$
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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)
