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Chapter 7: Problem 2

In the HCl molecule, the separation between the nuclei of the two atoms is about \(1.27 \mathrm{~A}\left(1 \mathrm{~A}=10^{-10} \mathrm{~m}\right) .\) Find the approximate location of the \(\mathrm{CM}\) of the molecule, given that a chlorine atom is about \(35.5\) times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.

Short Answer

Expert verified
The center of mass of the HCl molecule is approximately \(1.236 \times 10^{-10} \text{ m}\) from the hydrogen nucleus.

Step by step solution

01

Identify given data

We have two atoms: hydrogen (H) and chlorine (Cl). The separation between the nuclei is given as \(1.27 \text{ 脜}\). Chlorine is \(35.5\) times as massive as hydrogen.
02

Calculate masses of atoms

Let the mass of the hydrogen atom be \(m_H = m\). Therefore, the mass of the chlorine atom is \(m_{Cl} = 35.5m_H = 35.5m\).
03

Determine the position of the center of mass

The formula for the center of mass (CM) of a two-particle system along a line is \(x_{CM} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}\). Let the position of the hydrogen nucleus be \(x_H = 0\) and the position of the chlorine nucleus be \(x_{Cl} = 1.27 \times 10^{-10} \text{ m}\).
04

Substitute values into the CM formula

Substitute \(m_H = m\), \(x_H = 0\), \(m_{Cl} = 35.5m\), and \(x_{Cl} = 1.27 \times 10^{-10} \text{ m}\) into the formula: \[x_{CM} = \frac{m \times 0 + 35.5m \times 1.27 \times 10^{-10}}{m + 35.5m}\]
05

Simplify the expression

The expression becomes: \[x_{CM} = \frac{35.5m \times 1.27 \times 10^{-10}}{36.5m}\] The \(m\)'s cancel out: \[x_{CM} = \frac{35.5 \times 1.27 \times 10^{-10}}{36.5}\]
06

Compute the numeric value

Calculate \(x_{CM}\): \[x_{CM} \approx 1.236 \times 10^{-10} \text{ m}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the HCl Molecule
The HCl molecule is a simple diatomic molecule formed by hydrogen and chlorine atoms. It is made of one hydrogen atom (H) and one chlorine atom (Cl). This molecule is an example of a polar covalent bond. In this type of bond, electrons are not shared equally. Chlorine attracts the electrons more strongly than hydrogen because it is more electronegative. This property causes the HCl molecule to have a partial positive charge near the hydrogen and a partial negative charge near the chlorine.
  • **Diatomic molecule:** Composed of two atoms.
  • **Polar bond:** Electrons are closer to one atom than the other.
  • **Electronegativity difference:** Results in partial charges.
Polarity leads to unique physical and chemical properties, such as HCl being soluble in water and forming hydrochloric acid when dissolved. This molecule serves as a simple example to understand more complex molecular structures.
Grasping the Concept of Atomic Mass
The atomic mass of an element is a critical concept in understanding molecules like HCl. Atomic mass refers to the mass of an atom, which is concentrated in its nucleus, where the protons and neutrons reside. Electrons have a negligible mass compared to the nucleus.
  • **Protons and neutrons:** Determine the atomic mass.
  • **Electrons:** Have very little impact due to low mass.
In the HCl molecule, the chlorine atom is approximately 35.5 times more massive than the hydrogen atom. This difference is crucial in calculating the center of mass. Understanding how atomic mass varies among elements helps in various calculations and predictions about molecular behavior, including the physical properties of molecules.
Nucleus Separation in Molecules
The separation of nuclei in molecules is a vital factor in understanding molecular structure and bonding. In the HCl molecule, the distance between the nuclei of the hydrogen and chlorine atoms is given as 1.27 脜 (angstroms), which is equivalent to 1.27 x 10^-10 meters.
  • **Inter-nuclear distance:** Influences bond strength and molecule stability.
  • **Measurement unit:** 脜ngstr枚m (脜), commonly used for atomic scales.
The concept of nucleus separation is essential because it affects the energy and length of the bond in a molecule. Molecules tend to arrange themselves at a distance that minimizes energy, thus stabilizing the structure. Applying this concept allows chemists to predict molecular behavior and reactivity in chemical processes.

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Most popular questions from this chapter

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