Problem 55 - Physics

Question

Interactive Solution \(14.55\) 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 approximately 2810 meters.

Step 1: Calculate average translational kinetic energy

The average translational kinetic energy of a gas particle is given by the formula \( KE = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant \( 1.38 \times 10^{-23} \mathrm{~J/K} \) and \( T \) is the temperature in Kelvin. Substituting \( T = 291 \mathrm{~K} \), we get:\[KE = \frac{3}{2} \times 1.38 \times 10^{-23} \times 291 \approx 6.02 \times 10^{-21} \mathrm{~J}.\]

Step 2: Equate kinetic energy to gravitational potential energy

The kinetic energy converted to gravitational potential energy at the maximum height is given by \( mgh \), where \( m \) is the mass of the xenon atom, \( g \) is the acceleration due to gravity (\( 9.81 \mathrm{~m/s^2} \)), and \( h \) is the height. We set \( KE = mgh \).

Step 3: Convert atomic mass to kilograms

First, convert the atomic mass of xenon into kilograms. Given the atomic mass is \( 131.29 \mathrm{u} \) and \( 1 \mathrm{u} = 1.66 \times 10^{-27} \mathrm{~kg} \), the mass \( m \) is:\[m = 131.29 \times 1.66 \times 10^{-27} \approx 2.18 \times 10^{-25} \mathrm{~kg}.\]

Step 4: Solve for maximum height

Substitute \( m = 2.18 \times 10^{-25} \mathrm{~kg} \) into \( KE = mgh \), and solve for \( h \):\[6.02 \times 10^{-21} = 2.18 \times 10^{-25} \times 9.81 \times h.\]Rearranging gives:\[h = \frac{6.02 \times 10^{-21}}{2.18 \times 10^{-25} \times 9.81} \approx 2810 \mathrm{~m}.\]

Key Concepts

Translational Kinetic Energy

Translational kinetic energy is a form of energy related to the motion of particles. When discussing gases, this kind of energy refers to the energy found in the random movement of molecules or atoms in the gas phase. It’s crucial to understand that translational kinetic energy specifically deals with linear motion, unlike rotational or vibrational kinetic energies, which involve rotation and oscillation, respectively.

The average translational kinetic energy of an ideal gas particle is expressed with the formula \( KE = \frac{3}{2}kT \). Here, \( k \) stands for the Boltzmann constant, and \( T \) is the temperature in Kelvin. This equation indicates that the energy an atom or molecule possesses due to its motion is directly proportional to the temperature of the environment.

When calculating, for example, the energy of a xenon atom at 291 K, we substitute into the equation to find an energy value. This energy helps predict behaviors such as how high an atom might ascend in the gravitational field before its kinetic energy is entirely converted to potential energy.

Gravitational Potential Energy

Gravitational potential energy is the energy held by an object due to its position in a gravitational field. It’s a fundamental concept when trying to understand the energy transformations that happen as an object changes altitude, like an atom moving upward before coming to rest.

The formula for gravitational potential energy is \( mgh \), where \( m \) represents mass, \( g \) is the acceleration due to gravity (or gravitational force), and \( h \) indicates the height. This equation shows that potential energy depends on three factors: the mass of the object, the gravitational pull it experiences, and how high it is.

In exercises like the one with the xenon atom, gravitational potential energy plays a crucial role because it equals the kinetic energy when the object has reached its peak height. Understanding this transformation from kinetic to potential energy is key to solving energy conservation problems.

Boltzmann Constant

The Boltzmann constant is a fundamental physical constant that acts as a bridge between the macroscopic and microscopic worlds. Its value is \( 1.38 \times 10^{-23} \mathrm{~J/K} \). This constant is named after Ludwig Boltzmann, who made significant contributions to statistical mechanics, helping to understand how the microscopic properties of atoms can be used to ascertain the bulk properties of gases.

Whenever calculating the translational kinetic energy of particles in a gas, the Boltzmann constant comes into play as it relates the average kinetic energy per particle to the absolute temperature of the gas. The presence of this constant in the kinetic energy formula highlights the thermal agitation of the particles within a gas and shows that at a higher temperature, the particles move more energetically.

Atomic Mass Unit

The atomic mass unit (amu or u) is a standard unit of mass that quantifies mass on an atomic or molecular scale. One atomic mass unit is defined as precisely \( 1/12 \) of the mass of a carbon-12 atom, and in kilograms, it’s roughly equal to \( 1.66 \times 10^{-27} \mathrm{~kg} \).

Understanding atomic mass units is crucial when dealing with problems in physics and chemistry that require the conversion of atomic masses from amu to kilograms. For instance, when calculating the kinetic or potential energy of a single atom or molecule, knowing its mass in kilograms is necessary for using standard physics equations.

In the exercise with the xenon atom, the mass was given in atomic mass units and converted to kilograms to utilize formulas dealing with energy in terms of joules, thereby allowing accurate and practical calculations.