Question

At \(25^{\circ} \mathrm{C}\) (approximately room temperature) the 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?

Short answer

(a) The rms speed of an Ar atom at room temperature in meters per second is approximately \(431.39 \, \mathrm{m/s}\). (b) The kinetic energy of one Ar atom is \(K.E = \frac{1}{2}(m_{atom})(431.39 \, \mathrm{m/s})^2\). (c) The total kinetic energy of one mole of Ar atoms is \(K.E_{total} = K.E \times 6.022 \times 10^{23}\).

Step-by-step solution

Convert Velocity

First, we need to convert the given velocity from km/h to m/s. To do this, we use the following conversion factors: 1 km = 1000 meters (m) 1 hour = 3600 seconds (s) So, we can write the expression to convert the given velocity: \(v_{rms} = 1553 \cdot \frac{1000 \, \text{m}}{3600 \, \mathrm{s}}\)

Calculate Velocity in m/s

Now, calculate the new value for the rms velocity: \(v_{rms} = 1553 \cdot \frac{1000}{3600} = 431.39 \, \text{m/s}\) Thus, the rms speed of an Ar atom at room temperature in meters per second is approx. 431.39 m/s.

Calculate the Kinetic Energy of one Ar atom

We now need to calculate the kinetic energy of an Ar atom. We use the formula for kinetic energy: \(K.E = \frac{1}{2}mv^2\) where m is the mass of the Ar atom and v is the rms velocity. The atomic mass of Ar is about 39.95 g/mol. We need to first convert it to kg: \(m = \frac{39.95}{1000} = 0.03995 \, \mathrm{kg}\) Note that this mass corresponds to one mole of Ar atoms. To find the mass of a single atom, we can use Avogadro's number (6.022 × 10^23 atoms/mol): \(m_{atom} = \frac{0.03995 \, \mathrm{kg}}{6.022 \times 10^{23} \, \mathrm{atoms}}\) Now, we can plug in the value of the mass of an Ar atom and the rms velocity into the kinetic energy formula: \(K.E = \frac{1}{2}(m_{atom})(431.39 \, \mathrm{m/s})^2\)

Calculate the Kinetic Energy for one mole of Ar atoms

Lastly, to find the total kinetic energy of one mole of Ar atoms, we multiply the kinetic energy of a single atom by Avogadro's number: \(K.E_{total} = K.E \times 6.022 \times 10^{23}\) Now, calculate the values and find the kinetic energy of one Ar atom and one mole of Ar atoms.

Key concepts

rms velocity

The term 'rms velocity' stands for root mean square velocity, which is a measure of the speed of particles in a gas. It provides an average speed of the gas particles, taking into account all speeds that the particles in the sample can have.
The formula for rms velocity is given by:

• \(v_{rms} = \sqrt{\frac{3kT}{m}}\)

Here, \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of a gas particle.
Understanding rms velocity is crucial because it relates to the kinetic energy of the particles in the gas, which in turn influences physical properties like pressure and temperature.
In this exercise, we started by converting the given rms velocity from kilometers per hour to meters per second, a standard scientific unit, to make further calculations of kinetic energy possible. This kind of conversion is essential in science where consistency of units is crucial.

Avogadro's number

Avogadro's number, denoted as \(6.022 \times 10^{23}\), is a fundamental constant in chemistry. It is the number of constituent particles, usually atoms or molecules, that are contained in the amount of substance known as one mole.
Understanding Avogadro's number allows us to relate microscopic atomic and molecular scales to macroscopic amounts of material that we can measure directly.
In the context of kinetic energy calculations, as in the provided exercise, Avogadro's number is used to determine the mass of a single atom when the mass of a mole is known. This is critical since kinetic energy for an individual atom can then be calculated, which can then be easily scaled to find the energy for a whole mole of atoms.
This constant forms the bridge between the quantifiable world of laboratory chemistry and the invisible atomic world, making it a powerful tool for calculations at the molecular level.

mole concept

The mole concept is a core principle in chemistry that serves as a link between the mass of a material and the number of atoms or molecules it contains. Essentially, a mole is a unit of measure that corresponds to Avogadro's number of particles. It allows chemists to count particles by weighing them.
One mole of any substance contains exactly \(6.022 \times 10^{23}\) (Avogadro's number) of constituent particles, be they atoms, molecules, ions, etc. This concept is incredibly useful in various chemical calculations, including those of kinetic energy as shown in the exercise.
In calculating kinetic energy for a mole of argon atoms, the mole concept helps us by providing a clear pathway from atomic scale calculations (like those for an individual particle's velocity and energy) to amounts that we can practically work with in the laboratory. With molecular weights and Avogadro's number, we can determine how a given mass of substance correlates with its number of constituent particles, enabling precise and meaningful calculations.