Quantum Mechanics
Name (In Arabic):
Student Id.: $\qquad$
Hints:
$$
\begin{array}{ll}
h=6.62607 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} & \text { Speed of light }(c)=2.9979 \times 10^{4} \mathrm{~m} / \mathrm{s} \
\tilde{v}=1 / \lambda=v / c \quad v=\frac{1}{2 \pi} \sqrt{\frac{k}{\mu}} & E_{*}=h v\left(n+\frac{1}{2}\right)
\end{array}
$$

The strongest infrared band of ${ }^{12} \mathrm{C}^{1} \mathrm{H}$ occurs at $3270 \mathrm{~cm}^{-1}$.
a) Find the force constant of ${ }^{12} \mathrm{C}^{1} \mathrm{H}$.
b) The approximate zero-point energy of ${ }^{12} \mathrm{C}^{1} \mathrm{H}$

Question

Quantum Mechanics
Name (In Arabic):
Student Id.: $\qquad$
Hints:
$$
\begin{array}{ll}
h=6.62607 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} & \text { Speed of light }(c)=2.9979 \times 10^{4} \mathrm{~m} / \mathrm{s} \
\tilde{v}=1 / \lambda=v / c \quad v=\frac{1}{2 \pi} \sqrt{\frac{k}{\mu}} & E_{*}=h v\left(n+\frac{1}{2}\right)
\end{array}
$$

The strongest infrared band of ${ }^{12} \mathrm{C}^{1} \mathrm{H}$ occurs at $3270 \mathrm{~cm}^{-1}$.
a) Find the force constant of ${ }^{12} \mathrm{C}^{1} \mathrm{H}$.
b) The approximate zero-point energy of ${ }^{12} \mathrm{C}^{1} \mathrm{H}$

Accepted Answer

∻Solution∻ ‖ 1 ⋮ **Convert wavenumber to frequency**: The wavenumber $ ilde{ u}$ given in cm$^{-1}$ must be converted to frequency $ u$ in s$^{-1}$. This is done by multiplying the wavenumber by the speed of light $c$ in cm/s. ‖ ‖ $ u = ilde{ u} imes c = 3270 \, ext{cm}^{-1} imes 2.9979 imes 10^{10} \, ext{cm/s}$ ‖ ‖ 2 ⋮ **Calculate the reduced mass $\mu$**: The reduced mass $\mu$ of the diatomic molecule $^{12}C^1H$ is calculated using the formula $\mu = \frac{m_1 imes m_2}{m_1 + m_2}$, where $m_1$ and $m_2$ are the masses of the two atoms. The mass of $^{12}C$ is approximately 12 amu and the mass of $^1H$ is approximately 1 amu. ‖ ‖ $\mu = \frac{12 imes 1}{12 + 1} imes 1.66054 imes 10^{-27} \, ext{kg/amu} = 0.913 imes 1.66054 imes 10^{-27} \, ext{kg}$ ‖ ‖ 3 ⋮ **Find the force constant $k$**: Using the equation $ u = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}}$, solve for $k$. ‖ ‖ $k = (2\pi u)^2 imes \mu$ ‖ ‖ 4 ⋮ **Calculate the zero-point energy $E_0$**: The zero-point energy is given by $E_0 = h u\left(n + \frac{1}{2} ight)$ for $n=0$. ‖ ‖ $E_0 = h u\left(0 + \frac{1}{2} ight) = \frac{1}{2}h u$ ‖ ∻1 Answer∻ ⚹ [Insert force constant $k$ here] ⚹ ∻2 Answer∻ ⚹ [Insert zero-point energy $E_0$ here] ⚹ ∻Key Concept∻ ⚹ The force constant and zero-point energy of a diatomic molecule can be determined using quantum mechanics equations and the properties of the molecule. ⚹ ∻Explanation∻ ⚹ The force constant is related to the stiffness of the bond and can be calculated from the vibrational frequency, while the zero-point energy is the minimum energy that the molecule possesses even at absolute zero temperature. ⚹◊What is the formula to calculate the zero-point energy of a quantum harmonic oscillator?⍭ Generate me a similar question◊

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