Problem 140 - Chemistry

Question

The Chemistry in Action essay "Super Cold Atoms" in Section \(5.7 .\) describes the cooling of rubidium vapor to \(5.0 \times 10^{-8} \mathrm{~K}\). Calculate the root-mean-square speed and average kinetic energy of a \(\mathrm{Rb}\) atom at this temperature.

Step-by-Step Solution

Answer

The root-mean-square speed of a Rubidium atom at \(5.0 \times 10^{-8} \mathrm{~K}\) is \(0.12 m/s\) and the average kinetic energy is \(1.035 \times 10^{-29} J\).

Step 1: Calculate the Mass of a Rubiuim Atom

Rubidium (Rb) has a molar mass of 85 g/mol. But the mass of one atom is needed here. Using Avogadro's number (\(N_A = 6.022 \times 10^{23} \, atoms/mol\)), the mass of one rubidium atom can be calculated as \(m = M/N_A = 85g/mol / 6.022 \times 10^{23} atoms/mol = 1.41 \times 10^{-25} kg/atom\).

Step 2: Calculate the Root-Mean-Square Speed

With the mass of one Rb atom and the given temperature (T), substitute the values into the RMS speed equation \(v_{rms} = \sqrt{3kT/m}\) to obtain \(v_{rms} = \sqrt{3\cdot \, 1.38 \cdot 10^{-23} J/K \cdot 5 \cdot 10^{-8} K / 1.41 \cdot 10^{-25} kg} = 0.12 m/s.\)

Step 3: Calculate the Average Kinetic Energy

Substitute the given temperature (T) into the average kinetic energy equation \(KE_{avg} =\frac{3}{2} kT\) to obtain \(KE_{avg} =\frac{3}{2} \cdot 1.38 \cdot 10^{-23} J/K \cdot 5 \cdot 10^{-8} K = 1.035 \times 10^{-29} J\).

Key Concepts

Root-Mean-Square Speed

The root-mean-square speed (RMS speed) is a crucial concept when discussing the motion of particles within a gas. It provides a measure of the average velocity of the atoms or molecules within that gas. In physics and chemistry, these speeds relate to the kinetic energy and temperature of the particles.
The equation for the root-mean-square speed is given by:

• \( v_{rms} = \sqrt{\frac{3kT}{m}} \)

where:

• \( k \) is the Boltzmann constant, which is \( 1.38 \times 10^{-23} \) J/K.
• \( T \) is the temperature in Kelvin.
• \( m \) is the mass of a single particle in kilograms.

In the case of rubidium atoms cooled to ultra-low temperatures, the RMS speed becomes remarkably small. For instance, at \( 5.0 \times 10^{-8} \) K, rubidium's RMS speed drops to just \( 0.12 \) m/s. This demonstrates the intimate relationship between temperature and molecular speed.

Rubidium Atoms

Rubidium (Rb) is an element located in group 1 on the periodic table, known as an alkali metal. It is characterized by its soft, silvery appearance and high reactivity, similar to other elements in this group like sodium and potassium.
Rubidium atoms have a significant role in experiments involving ultra-cold physics, particularly in areas like Bose-Einstein condensates. When cooled to near absolute zero, rubidium gases can exhibit unique quantum phenomena.

• With a molar mass of \( 85 \) g/mol, rubidium helps illustrate key physics concepts like kinetic energy and root-mean-square speed.
• Understanding rubidium's behavior at minimal temperatures aids in comprehending how atomic and sub-atomic particles perform under extreme conditions.

Applying these principles to rubidium atoms allows scientists to explore atomic interactions that are otherwise hidden at higher temperatures.

Temperature Dependence

Temperature plays a vital role in determining the kinetic properties of atoms and molecules. As temperature increases, the kinetic energy of particles also increases, which in turn affects the root-mean-square speed. The link between temperature and kinetic energy can be expressed through:

• The kinetic energy equation \( KE_{avg} = \frac{3}{2}kT \)

Here, \( KE_{avg} \) represents the average kinetic energy per particle.
In ultra-cold conditions like those for rubidium vapor at \( 5.0 \times 10^{-8} \) K, the kinetic energy is exceptionally low. Such extreme cooling causes the particles to move very slowly, which impacts their speed and interactions.

• This dependency on temperature is essential for exploring fundamental physics and quantum mechanics.
• By manipulating temperature, scientists can control the state, behavior, and properties of materials and particles.

Molar Mass

Molar mass is a measure of the mass of a given substance, expressed in grams, divided by the amount of substance measured in moles. It is crucial for converting between mass and the number of moles, facilitating calculations in both chemistry and physics.
The molar mass of rubidium is \( 85 \) g/mol, which is necessary for determining the mass of a single rubidium atom. To find the mass of one atom:

• Use Avogadro's number, \( 6.022 \times 10^{23} \) atoms/mol.
• Divide the molar mass by Avogadro's number.
• For rubidium, this yields a single atom mass of \( 1.41 \times 10^{-25} \) kg/atom.

Understanding molar mass helps in various calculations, such as in determining root-mean-square speed and average kinetic energy, highlighting its importance in scientific studies.