Problem 55 - Physics

Question

Interactive Solution \(\underline{14.55}\) at provides a model for problems of this type. The temperature near the surface of the earth is \(291 \mathrm{~K}\). A xenon atom (atomic mass \(=131.29 \mathrm{u}\) ) has a kinetic energy equal to the average translational kinetic energy and is moving straight up. If the atom does not collide with any other atoms or molecules, how high up would it go before coming to rest? Assume that the acceleration due to gravity is constant throughout the ascent.

Step-by-Step Solution

Answer

The xenon atom would rise to approximately 28100 meters before coming to rest.

Step 1: Determine the Average Translational Kinetic Energy

The average translational kinetic energy of a gas particle can be given by the equation \( KE_{avg} = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant \( 1.38 \times 10^{-23} \, \text{J/K} \) and \( T \) is the temperature in Kelvin. Here, \( T = 291 \, \text{K} \).Calculate the average translational kinetic energy:\[ KE_{avg} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 291 = 6.01 \times 10^{-21} \, \text{J} \]

Step 2: Relate Kinetic Energy to Potential Energy

When the gas particle climbs to its highest point, all its initial kinetic energy will be converted into gravitational potential energy (GPE). The potential energy \( PE \) is given by \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity \( 9.8 \, \text{m/s}^2 \), and \( h \) is the height.Since \( KE_{avg} = PE \), we have:\[ 6.01 \times 10^{-21} = m \times 9.8 \times h \]

Step 3: Calculate the Mass of a Xenon Atom

The mass \( m \) of a xenon atom can be calculated using its atomic mass \( 131.29 \mathrm{u} \) and converting it to kilograms. 1 atomic mass unit \( \mathrm{u} \) is equal to \( 1.66 \times 10^{-27} \, \text{kg} \).\[ m = 131.29 \times 1.66 \times 10^{-27} = 2.18 \times 10^{-25} \, \text{kg} \]

Step 4: Solve for Height

Substitute the mass into the equation for potential energy:\[ 6.01 \times 10^{-21} = 2.18 \times 10^{-25} \times 9.8 \times h \]Solve for \( h \):\[ h = \frac{6.01 \times 10^{-21}}{2.18 \times 10^{-25} \times 9.8} = 2.81 \times 10^4 \, \text{m} \]

Step 5: Conclusion

The final height that the xenon atom would reach before coming to a stop is approximately \( 2.81 \times 10^4 \, \text{m} \).

Key Concepts

Potential Energy

Potential energy is the energy possessed by an object due to its position or state. For example, an object at a height in a gravitational field has potential energy because gravity can act on it to move it to a lower height. In our scenario, when the xenon atom travels upward, it gains gravitational potential energy. This is because it moves against the gravitational pull of the Earth. As it moves higher, the potential energy increases, while its kinetic energy decreases until the atom comes to rest.
To calculate potential energy, use the formula:

• \( PE = mgh \)
  
• where \( m \) is the mass of the xenon atom, \( g \) is the acceleration due to gravity, and \( h \) is the height.

This formula shows that potential energy increases with height and mass.

Kinetic Energy

Kinetic energy is the energy an object has due to its motion. It depends on the mass of the object and its velocity. In this problem, the xenon atom has kinetic energy because it moves upwards through the air. Initially, its kinetic energy is at the maximum as it starts its journey upwards from the Earth's surface.
For gases, the average translational kinetic energy is related to temperature using the formula:

• \( KE_{avg} = \frac{3}{2}kT \)
  
• where \( k \) is the Boltzmann constant and \( T \) is the temperature.

As the xenon atom rises, its kinetic energy decreases, converting into potential energy until the atom halts at its highest point.

Gravitational Potential Energy

Gravitational potential energy is a specific type of potential energy related to an object's position in a gravitational field. Any object, including the xenon atom, that rises in this field gains gravitational potential energy because it moves further from the Earth's center.
In this exercise, as the xenon atom ascends without any collisions, its kinetic energy completely transforms into gravitational potential energy at its peak height. This energy change illustrates the conservation of energy principle — energy is never lost, only transformed from one type to another.
Gravitational potential energy can be described with the formula:

• \( PE = mgh \), where
  
• \( m \) is the mass of the object, \( g \) is the gravitational constant, and \( h \) is the height above the reference point.

The transformation from kinetic to gravitational potential energy is crucial for solving how high the atom will travel.