Question

A nitrogen molecule ( \(\mathrm{N}_{2}\) ) having the average kinetic energy at \(300 \mathrm{K}\) is released from Earth's surface to travel upward. If the molecule could move upward without colliding with other molecules, then how high would it go before coming to rest? Give your answer in kilometers. [Hint: When the molecule comes to rest, the potential energy of the molecule will be \(m g h\) where \(m\) is the molecular mass in kilograms, \(g=9.81 \mathrm{m} \mathrm{s}^{-2}\) is the acceleration due to gravity, and \(h\) is the height, in meters, above Earth's surface.]

Short answer

The nitrogen molecule would reach an elevation of approximately 90.5 kilometers above Earth's surface before coming to rest.

Step-by-step solution

Calculate Initial Kinetic Energy of Nitrogen Molecule

First, calculate the initial kinetic energy of the nitrogen molecule. The average kinetic energy at \(300K\) can be given by \(kT\), where \(k = 1.38 x 10^{-23} J/K\) is the Boltzmann constant and \(T = 300K\) is temperature. The kinetic energy is then calculated by multiplying the given temperature with the Boltzmann constant, i.e., \(Ek_{initial} = kT = 1.38 x 10^{-23} J/K x 300 K = 4.14 x 10^{-21} J\).

Calculate Molecular Mass

As nitrogen (N) has an atomic mass of approximately 14 atomic mass units (a.m.u), the molecular mass (as \(\mathrm{N}_{2}\)) is \(2x14 = 28 a.m.u\). Convert this to kilograms by realizing that \(1 a.m.u = 1.66 x 10^{-27} Kg\), so \(m = 28 a.m.u * 1.66 x 10^{-27} Kg/a.m.u = 4.65 x 10^{-26} Kg\).

Calculate Final Potential Energy

When the nitrogen molecule comes to rest at height \(h\), it will have potential energy given by \(mgh = Ek_{initial}\), where \(g\) is acceleration due to gravity \(9.81 m/s^{2}\).

Solve for Height

Now, solve the previous equation for \(h\). Rearranging the equation with respect to \(h\), we have \(h = Ek_{Initial}/(mg) = 4.14 x 10^{-21} J / (4.65 x 10^{-26} Kg x 9.81 m/s^{2}) = 90473 m\).

Key concepts

Nitrogen Molecule

Nitrogen is the most abundant gas in Earth's atmosphere, making up about 78% by volume. A nitrogen molecule, denoted as \( \mathrm{N}_{2} \), consists of two nitrogen atoms bonded together. This diatomic molecule is quite stable due to a strong triple bond between the atoms.
Understanding the behavior of \( \mathrm{N}_{2} \) molecules, especially under various physical conditions, such as changes in energy, is important in both chemistry and physics.
One key aspect of molecular behavior is its kinetic energy. At any given temperature, the molecules of nitrogen possess a certain amount of kinetic energy. This energy is a consequence of their motion, with higher temperatures resulting in greater energy levels.

• At a temperature of \( 300 \mathrm{K} \), the kinetic energy can be computed using the formula \( kT \) (Boltzmann constant \( k \times \) temperature \( T \)).
• The Boltzmann constant is approximately \( 1.38 \times 10^{-23} \) Joules per Kelvin (\( J/K \)).

Through understanding these properties, we get an insight into how molecules collide and interact, which is essential in various scientific fields.

Potential Energy

Potential energy is the stored energy of an object due to its position or configuration. For example, when an object is lifted in a gravitational field, it gains potential energy relative to its position at ground level.
In the context of a nitrogen molecule rising away from the Earth's surface, potential energy plays a crucial role. As the molecule travels upward without external interference, it slows down and eventually stops due to the conversion of its kinetic energy into potential energy.

• The potential energy (\( mgh \)) is dependent on:
• The mass \( m \) of the nitrogen molecule, calculated from its atomic mass on the molecular scale.
• The gravitational acceleration \( g \), which is approximately \( 9.81 \mathrm{m/s}^{2} \).
• Height \( h \), which is the altitude reached above Earth's surface.

Thus, at the height where all kinetic energy is converted to potential energy, the nitrogen molecule comes to rest. This balance of energies showcases the fundamental principles of work and energy conservation.

Gravitational Force

Gravitational force is a natural phenomenon by which all things with mass are attracted to one another, including the nitrogen molecule on Earth.
This force is directly related to the masses involved and inversely proportional to the square of the separation distance between the centers of mass.

• It makes objects fall toward Earth, providing the weight of the objects. For our purpose, it helps us understand the interaction between the nitrogen molecule and Earth's gravity.
• The gravitational force accelerates objects at \( 9.81 \mathrm{m/s}^{2} \) near the surface of the Earth.

In the nitrogen molecule exercise, we factor in gravitational force to calculate how far a nitrogen molecule would travel before its kinetic energy is entirely transformed into potential energy upon reaching a certain height. This scenario exemplifies the pull of gravity and its effect on motion and energy conversion.