Problem 118 - Chemistry The Central Science In Si Units

Question

At \(25^{\circ} \mathrm{C}\) (approximately room temperature) the \(\mathrm{rms}\) velocity of an Ar atom in air is \(1553 \mathrm{~km} / \mathrm{h}\). (a) What is the rms speed in \(\mathrm{m} / \mathrm{s}\) ? (b) What is the kinetic energy (in J) of an Ar atom moving at this speed? (c) What is the total kinetic energy of \(1 \mathrm{~mol}\) of Ar atoms moving at this speed?

Step-by-Step Solution

Answer

(a) 431.4 m/s; (b) \(6.17 \times 10^{-21}\) J; (c) 3711 J for 1 mole.

Step 1: Convert rms velocity units

We need to convert the rms velocity of an Ar atom from kilometers per hour to meters per second. Given: \( 1553 \mathrm{~km/h} \). The conversion factors required are: \( 1 \mathrm{~km} = 1000 \mathrm{~m} \) and \( 1 \mathrm{~hour} = 3600 \mathrm{~s} \).So, \( 1553 \mathrm{~km/h} \times \frac{1000 \mathrm{~m}}{1 \mathrm{~km}} \times \frac{1 \mathrm{~hour}}{3600 \mathrm{~s}} = \frac{1553 \times 1000}{3600} \mathrm{~m/s} \). This results in approximately \( 431.4 \mathrm{~m/s} \).

Step 2: Calculate the kinetic energy of one Ar atom

The equation for the kinetic energy of a particle is \( E_k = \frac{1}{2} m v_{rms}^2 \), where \( m \) is the mass of the Ar atom and \( v_{rms} = 431.4 \mathrm{~m/s} \). 1 mole of Ar atoms weighs \( 39.948 \mathrm{~g} \) or \( 0.039948 \mathrm{~kg} \). Thus, the mass \( m \) of one Ar atom is \( \frac{0.039948}{6.022 \times 10^{23}} \mathrm{~kg} \). Using \( m \approx 6.64 \times 10^{-26} \mathrm{~kg} \), the kinetic energy is:\[ E_k = \frac{1}{2} \times 6.64 \times 10^{-26} \times (431.4)^2 \approx 6.17 \times 10^{-21} \mathrm{~J} \].

Step 3: Calculate kinetic energy for 1 mole of Ar atoms

To find the total kinetic energy for 1 mole of Ar atoms, we multiply the kinetic energy of one atom by Avogadro's number: \( 6.022 \times 10^{23} \). Thus, the total kinetic energy for 1 mole is:\[ 6.17 \times 10^{-21} \mathrm{~J/atom} \times 6.022 \times 10^{23} \approx 3711 \mathrm{~J} \].

Key Concepts

RMS Speed

RMS Speed stands for Root Mean Square Speed, an important concept in physics, especially in molecular kinetics. It represents the speed of particles in a gas that is most comparable to the average speed. The RMS speed is derived from the kinetic theory of gases, which assumes that gases consist of many small particles that are in constant, random motion.

In calculations, RMS speed is essential for determining other physical properties, such as kinetic energy. The formula for RMS speed is:

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

where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single molecule.
This helps us understand how fast the gas particles move at a given temperature and mass.

Molecular Kinetics

Molecular kinetics, also known as kinetic theory of gases, offers insights into how gas molecules behave. It uses several assumptions to describe the motion of particles in a gas, such as assuming elastic collisions and negligible volume.

In this theory:

• Particles are constantly moving and collide with each other and the walls of their container, exerting pressure.
• The temperature of a gas is proportional to the average kinetic energy of its particles, which is why temperature increases lead to faster molecular speeds.
• Each molecule moves randomly, but an average speed can still be determined through statistical means, like RMS speed.

These principles help in calculating properties like pressure, volume, and temperature and connecting them to the microscopic behavior of molecules.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry and physics that represents the number of particles in one mole of a substance, which is approximately \(6.022 \times 10^{23}\).

This number allows scientists to:

• Convert between moles and number of atoms or molecules, providing a bridge between macroscopic and microscopic scales.
• Calculate the mass of a single atom or molecule when the molar mass is known.
• Determine the total energy of a sample by multiplying the energy per particle by Avogadro's Number.

In our context, we use Avogadro's Number to find the total kinetic energy of a mole of gas particles once we know the energy for a single particle.

Unit Conversion

Unit conversion is crucial in physics and chemistry to ensure consistency and accuracy in measurements and calculations. It involves changing a quantity expressed in one set of units to another equivalent set.

Common unit conversions include:

• Distance: e.g., converting kilometers to meters, as 1 kilometer is equivalent to 1000 meters.
• Time: e.g., converting hours to seconds, as 1 hour equals 3600 seconds.
• Mass and volume conversions using molar mass and density when needed.

Understanding unit conversions allows us to compute accurately in different scenarios and is essential in achieving correct and meaningful results.

Molar Mass

Molar Mass is the mass of a given substance (chemical element or chemical compound) divided by its amount of substance. Its unit is grams per mole (g/mol).

In terms of calculations:

• The molar mass of an element is taken from its atomic mass found on the periodic table, for instance, Argon has a molar mass of approximately 39.948 g/mol.
• It allows for the calculation of the mass of single atoms by using Avogadro’s Number to determine the mass contribution of individual particles.
• It is utilized to convert between the mass of a substance and the amount in moles, thus bridging the gap between particles and macroscopic quantities.

Knowing the molar mass is essential for calculations involving chemical reactions, stoichiometry, and energy computations in physics and chemistry.