Question

In HCl molecule the separation between the nuclei of the two atoms is about \(1.27 \mathrm{~A}\left(1 \mathrm{~A}=10^{-10}\right)\). The approximate location of the centre of mass of the molecule is \(-\mathrm{A}\) i \(\wedge\) with respect of Hydrogen atom (mass of CL is \(35.5\) times of mass of Hydrogen \()\) \(\\{\mathrm{A}\\} 1 \mathrm{i}\) \\{B \\} \(2.5 \mathrm{i}\) \\{C \(\\} 1.24 \mathrm{i}\) \\{D \(1.5 \mathrm{i}\)

Short answer

The position of the center of mass of the HCl molecule with respect to the hydrogen atom is approximately \(1.24 Å\), which corresponds to option C.

Step-by-step solution

Understanding the center of mass formula

The center of mass (CM) of a two-particle system can be found using the following formula: CM \(=\frac{m_1x_1 + m_2x_2}{m_1 + m_2}\) where: - m1 and m2 are the masses of the two particles, - x1 and x2 are the positions of the two particles, and - CM is the position of the center of mass. In this case, we have only one dimension (the x-axis), so the formula can be simplified to: CM \(= \frac{m_Hx_H + m_{Cl}x_{Cl}}{m_H + m_{Cl}}\) where: - \(m_H\) is the mass of the hydrogen atom, - \(m_{Cl}\) is the mass of the chlorine atom, - \(x_H\) is the position of the hydrogen atom, and - \(x_{Cl}\) is the position of the chlorine atom.

Assigning values to the variables

Since we are calculating the position of the center of mass with respect to the hydrogen atom, we can set the position of hydrogen atom as the origin. Thus, \(x_H = 0\). Now, we only need to find the position of the chlorine atom (\(x_{Cl}\)) and the ratio of the masses of chlorine and hydrogen atoms (\(m_H\) and \(m_{Cl}\)) Given, the separation between the nuclei of the two atoms is: \(x_{Cl} = 1.27 Å\) Since \(1 Å = 10^{-10}\), the separation between the nuclei of the two atoms is: \(x_{Cl} = 1.27 * 10^{-10} meters\) Also, given that the mass of the chlorine atom is 35.5 times the mass of the hydrogen atom: \(m_{Cl} = 35.5 * m_H\)

Calculating the position of the center of mass

Now, we can use the formula for the center of mass to find the position of the center of mass: CM \(= \frac{m_H*0 + (35.5*m_H)(1.27*10^{-10})}{m_H + 35.5*m_H}\) As we can observe, the mass of the hydrogen atom (\(m_H\)) cancels out in the numerator and denominator: CM \(= \frac{(35.5)(1.27*10^{-10})}{1 + 35.5}\) On calculating the position of the center of mass, we get: CM \(= 1.2444 * 10^{-10} meters\)

Converting the center of mass position to Å and comparing to given options

Now, we convert the center of mass position back to Å: CM \(= 1.2444 * 10^{-10} meters * \frac{1 Å}{10^{-10} meters}\) CM \(= 1.2444 Å\) Comparing this result with the given options: A) \(1 Å\) B) \(2.5 Å\) C) \(1.24 Å\) D) \(1.5 Å\) The correct choice is option C) \(1.24 Å\), as it is the closest to our calculated result of 1.2444 Å.

Key concepts

Molecular Physics

Molecular physics delves into the study of molecules and the bonds between them. It's an area of physics that explores

• how atoms combine to form molecules
• how molecular interactions define the properties of materials
• the energy levels within molecules

In the context of molecular physics, understanding phenomena like the center of mass is crucial. The center of mass helps describe how molecules move as a whole. It’s a conceptual tool applied often to determine properties like motion and structure. Observing how the weight distributes in a molecule can help predict its behavior in physical processes. The center of mass offers an average position where all distributed mass is thought to be concentrated, easing calculations in molecular dynamics.

HCl Molecule

Hydrochloric acid (HCl) is a compound consisting of hydrogen (H) and chlorine (Cl) atoms. It serves as a simple example of a diatomic molecule in molecular physics studies.
The distance between the nuclei of the hydrogen and chlorine atoms is around 1.27 Å, which is a fraction of a nanometer: \((1 ext{ Å} = 10^{-10} ext{ meters})\). This relatively short distance highlights the bond strength between hydrogen and chlorine.
The HCl molecule offers a classic example in teaching center of mass concepts due to its simplicity. Here, understanding where the center of mass lies between the two atoms is crucial in various chemical reactions and properties assessments. The center of mass relation with such small-scale distances requires precise calculations but offers insightful applications across chemistry and molecular physics.

Atomic Mass Ratio

In the scenario of an HCl molecule, the atomic mass ratio between the chlorine and hydrogen atoms is crucial. Chlorine is noted to be \(35.5\) times heavier than hydrogen.

Why does this matter? Well, the atomic mass ratio helps predict where the center of mass will be. Chlorine’s significantly higher mass means the center of mass will be closer to the chlorine atom rather than midway between the hydrogen and chlorine atoms.

• Understanding mass differences helps in predicting molecular motion.
• Massively affects how molecules like HCl interact physically and chemically.
• Provides insights into the gravitational interactions at a molecular level.

In this specific problem, this understanding helps set up calculations correctly so you can solve for the center of mass accurately.

Coordinate System

A coordinate system provides a framework for locating points in space. In the case of molecular physics exercises like determining the center of mass of HCl, we usually assign a one-dimensional coordinate system across the separation between the molecules.
Positioning the hydrogen atom ( H ) at the origin simplifies calculations, grounding our reference point. From there, assessing the position of the chlorine atom requires merely measuring along the axis. Having a clear basis makes solving for molecular properties like the CM straightforward. In cases of multiple-dimensional forces being analyzed, you might extend the coordinate system to axes that account for further spatial dimensions.