7) The Pure Rotational spectrum of isotopically pure ${ }^{12} \mathrm{C}={ }^{25} \mathrm{O}$ at room temperature is shown at right. The $x$-axis numbers identify the values for $J^{2}$ for each transition.
a) If $B_{0}=5.763573^{*} 10^{\circ 0} \mathrm{~s}^{-1}$, and $T=115.4^{\circ} \mathrm{C}$, for which $\mathrm{I}$ value will you see the most intense peak? Assume the Rigid Rotor model applies.
b) Rotation+Vibration: Transition energies for the RoVib spectrum of ${ }^{12} \mathrm{C}={ }^{-11} \mathrm{O}$ are given In the table. Draw arrows onto the chart below and label with the letters from the table that show all the transitions.
(c) Rotation+Vibration, continued: Choose one transition from each of the $(R)$ and $(P)$ branches in the table and use the relation $\Delta E\left(\mathrm{~cm}^{-1}\right)=2 \bar{B}_{1}\left(2 \mathrm{~J}^{\prime \prime}+1\right)$ to calculate the rotational constant for the $\mathrm{v}=1$ state, $\overline{\mathrm{B}}_{1}$. Your answer should be in $\mathrm{cm}^{-1}$. Prove, in one mathematical equation, why $B_{1}

Q: 7) The Pure Rotational spectrum of isotopically pure ${ }^{12} \mathrm{C}={ }^{25} \mathrm{O}$ at room temperature is shown at right. The $x$-axis numbers identify the values for $J^{2}$ for each transition. a) If $B_{0}=5.763573^{*} 10^{\circ 0} \mathrm{~s}^{-1}$, and $T=115.4^{\circ} \mathrm{C}$, for which $\mathrm{I}$ value will you see the most intense peak? Assume the Rigid Rotor model applies. b) Rotation+Vibration: Transition energies for the RoVib spectrum of ${ }^{12} \mathrm{C}={ }^{-11} \mathrm{O}$ are given In the table. Draw arrows onto the chart below and label with the letters from the table that show all the transitions. (c) Rotation+Vibration, continued: Choose one transition from each of the $(R)$ and $(P)$ branches in the table and use the relation $\Delta E\left(\mathrm{~cm}^{-1}\right)=2 \bar{B}_{1}\left(2 \mathrm{~J}^{\prime \prime}+1\right)$ to calculate the rotational constant for the $\mathrm{v}=1$ state, $\overline{\mathrm{B}}_{1}$. Your answer should be in $\mathrm{cm}^{-1}$. Prove, in one mathematical equation, why $B_{1}<B_{0}$.

***This is a placeholder response as the actual Chemistry question requires specific data and calculations that are not provided in the text.*** ∻Key Concept∻ ⚹Boltzmann Distribution⚹ ∻Explanation∻ ⚹The most intense peak in a rotational spectrum corresponds to the most populated rotational level, which can be determined using the Boltzmann distribution at a given temperature.⚹ ∻Key Concept∻ ⚹Rotational-Vibrational Transitions⚹ ∻Explanation∻ ⚹Transitions between vibrational states involve changes in the rotational quantum number. R-branch transitions increase the rotational quantum number (ΔJ = +1), while P-branch transitions decrease it (ΔJ = -1).⚹ ∻Key Concept∻ ⚹Rotational Constant Calculation⚹ ∻Explanation∻ ⚹The rotational constant for the v=1 state can be calculated using the wavenumber values from the R and P branches and the given relation ΔE = 2Bv(2J"+1).⚹ ∻Key Concept∻ ⚹Rotational Constant Comparison⚹ ∻Explanation∻ ⚹Bv < B0 can be proven mathematically by considering the effects of anharmonicity on the rotational levels in different vibrational states. Anharmonicity causes a decrease in the rotational constant with increasing vibrational quantum number.⚹◊What mathematical equation can be used to calculate the rotational constant ($Bv$) for the $v=1$ state in part (c) of the problem⍭ Generate me a similar question◊

Question

7) The Pure Rotational spectrum of isotopically pure ${ }^{12} \mathrm{C}={ }^{25} \mathrm{O}$ at room temperature is shown at right. The $x$-axis numbers identify the values for $J^{2}$ for each transition.
a) If $B_{0}=5.763573^{*} 10^{\circ 0} \mathrm{~s}^{-1}$, and $T=115.4^{\circ} \mathrm{C}$, for which $\mathrm{I}$ value will you see the most intense peak? Assume the Rigid Rotor model applies.
b) Rotation+Vibration: Transition energies for the RoVib spectrum of ${ }^{12} \mathrm{C}={ }^{-11} \mathrm{O}$ are given In the table. Draw arrows onto the chart below and label with the letters from the table that show all the transitions.
(c) Rotation+Vibration, continued: Choose one transition from each of the $(R)$ and $(P)$ branches in the table and use the relation $\Delta E\left(\mathrm{~cm}^{-1}\right)=2 \bar{B}_{1}\left(2 \mathrm{~J}^{\prime \prime}+1\right)$ to calculate the rotational constant for the $\mathrm{v}=1$ state, $\overline{\mathrm{B}}_{1}$. Your answer should be in $\mathrm{cm}^{-1}$. Prove, in one mathematical equation, why $B_{1}<B_{0}$.

Accepted Answer

***This is a placeholder response as the actual Chemistry question requires specific data and calculations that are not provided in the text.*** ∻Key Concept∻ ⚹Boltzmann Distribution⚹ ∻Explanation∻ ⚹The most intense peak in a rotational spectrum corresponds to the most populated rotational level, which can be determined using the Boltzmann distribution at a given temperature.⚹ ∻Key Concept∻ ⚹Rotational-Vibrational Transitions⚹ ∻Explanation∻ ⚹Transitions between vibrational states involve changes in the rotational quantum number. R-branch transitions increase the rotational quantum number (ΔJ = +1), while P-branch transitions decrease it (ΔJ = -1).⚹ ∻Key Concept∻ ⚹Rotational Constant Calculation⚹ ∻Explanation∻ ⚹The rotational constant for the v=1 state can be calculated using the wavenumber values from the R and P branches and the given relation ΔE = 2Bv(2J"+1).⚹ ∻Key Concept∻ ⚹Rotational Constant Comparison⚹ ∻Explanation∻ ⚹Bv < B0 can be proven mathematically by considering the effects of anharmonicity on the rotational levels in different vibrational states. Anharmonicity causes a decrease in the rotational constant with increasing vibrational quantum number.⚹◊What mathematical equation can be used to calculate the rotational constant ($Bv$) for the $v=1$ state in part (c) of the problem⍭ Generate me a similar question◊

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