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Chapter 13: Problem 7

Carbon-monoxide. For a Carbon-monoxide molecule, the mass of the Carbon atom is \(12.0107 \mathrm{~g} / \mathrm{mol}\), and the mass of the Oxygen atom is \(15.9994 \mathrm{~g} / \mathrm{mol}\), and the typical distance from the Carbon to the Oxygen is \(112.8 \mathrm{pm}\). (a) Find the center of mass of a Carbon-monoxide molecule.

Short Answer

Expert verified
The center of mass is approximately \(64.47 \text{ pm}\) from the Carbon atom.

Step by step solution

01

- Identify the masses

The masses of the Carbon and Oxygen atoms are given. Specifically, the mass of the Carbon atom is given as \(12.0107\, \text{g/mol}\) and the mass of the Oxygen atom is \(15.9994\, \text{g/mol}\).
02

- Identify the distance

The distance between the Carbon and Oxygen atoms in the molecule is given as \(112.8\, \text{pm}\) or \(1.128 \times 10^{-10}\, \text{m}\).
03

- Center of mass formula

The center of mass \(x_{\text{cm}}\) of a two-particle system is given by the formula: \[ x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \] where \(m_1\) and \(x_1\) are the mass and position of the first particle (Carbon), and \(m_2\) and \(x_2\) are the mass and position of the second particle (Oxygen).
04

- Set coordinate positions

Assume the Carbon atom is at position \(x_1 = 0\) and the Oxygen atom is at position \(x_2 = 112.8 \text{pm}\).
05

- Substitute the values

Substitute the values into the center of mass formula: \[ x_{\text{cm}} = \frac{(12.0107 \times 0) + (15.9994 \times 112.8)}{12.0107 + 15.9994} \]
06

- Simplify the equation

Calculate the numerator and the denominator: \[ x_{\text{cm}} = \frac{0 + 1805.28}{28.0101} \]
07

- Final calculation

Complete the calculation to find the center of mass: \[ x_{\text{cm}} = \frac{1805.28}{28.0101} \text{ pm} \approx 64.47 \text{ pm} \]

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Key Concepts

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

two-particle system
A two-particle system is a simple and fundamental concept in physics and chemistry. It involves examining a system that contains exactly two particles or entities. In our case, the two particles are the Carbon and Oxygen atoms in a Carbon-monoxide (CO) molecule.
When we talk about such systems in relation to the center of mass, we apply specific principles to determine where the collective mass of these particles is concentrated. This is crucial because it helps in understanding various properties like motion, stability, and interaction with external forces.
  • Mass of the Particles: Each of the two particles has a specific mass which is essential for calculations.
  • Distance Between Particles: The separation between these particles affects where the center of mass lies.
By treating the CO molecule as a two-particle system, calculations become easier and more manageable.
mass of atoms
Atoms have different masses, and this variance plays a significant role in the behavior of molecules. In the case of Carbon-monoxide, the mass of the Carbon atom is approximately 12.0107 grams per mole (g/mol), while the mass of the Oxygen atom is about 15.9994 g/mol.
Understanding the mass of each atom is crucial for calculating the center of mass. The mass determines how each atom contributes to the molecule's overall mass distribution.
  • Importance of Atomic Mass: It directly influences the position of the center of mass.
  • Relative Mass: In CO, Oxygen is heavier than Carbon, which shifts the center of mass closer to the Oxygen atom.
By knowing the mass of the atoms, we can apply these values in formulas to find critical properties of the molecule.
molecular distance
Molecular distance refers to how far apart the atoms in a molecule are from each other. In our example, the distance between the Carbon and Oxygen atoms in a Carbon-monoxide molecule is 112.8 picometers (pm).
This distance is critical in determining the center of mass. It helps us understand where the center of mass will lie between these two atoms.
  • Measurement Units: Distance can be measured in various units like picometers, nanometers, meters, etc. Here, we use picometers.
  • Importance for Calculations: This distance is directly used in the formula for finding the center of mass. Knowing it helps in accurately determining the spatial properties of the molecule.
Accurate knowledge of molecular distance is essential for precise scientific calculations and understanding molecular behavior.

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Most popular questions from this chapter

Two-particle system. A \(2 \mathrm{~kg}\) particle is placed at \(x=2 \mathrm{~m}\) and a \(4 \mathrm{~kg}\) particle is placed at \(x=6 \mathrm{~m}\). (a) Where is the center of mass of this two-particle system?

Tetrahedron. A tetrahedron consists of four points (vertexes) connected by six lines of equal length, with three lines originating from each vertex, as seen in the figure. A particle of mass \(m\) is placed on each vertex. The coordinates of the corners are \((1,1,1),(-1,-1,1),(-1,1,-1),(1,-1,-1)\). (a) What is the center of mass of this system? (b) How does the center of mass change if we double the mass of the first particle?

Person in a boat. John (80 kg) is standing in a \(200 \mathrm{~kg}\) boat. He starts at one end and walks \(6 \mathrm{~m}\) to the other end of the boat. You may neglect drag forces between the boat and the water. (a) How far does the boat move in this process?

Car on a train. A \(1000 \mathrm{~kg}\) car is standing on an inclined plane on top of a \(2000 \mathrm{~kg}\) train cart. The cart rolls without friction on the track. The plane has an inclination of \(30^{\circ}\) with the horizontal. The car drives a horizontal distance of \(10 \mathrm{~m}\) from one end of the cart to the other. (a) How far has the cart moved in this process?
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