Question

Compare the root-mean-square speeds of \(\mathrm{O}_{2}\) and \(\mathrm{UF}_{6}\) at \(65^{\circ} \mathrm{C}\) .

Short answer

Gases with lower molecular masses have higher root mean square speeds at the same temperature, as the speed is inversely proportional to the square root of the molecular mass. So, at \(65^{\circ} C\), \(O_2\) molecules on average move faster than \(UF_6\) molecules.

Step-by-step solution

Identify Given Data

The given data are the temperature which is 65 degrees Celsius and the gases which are Oxygen \(O_2\) and Uranium hexafluoride \(UF_6\).

Convert Temperature to Kelvin

The temperature given is in Celsius. The formula for root mean square speed requires the temperature to be in Kelvin. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. Thus, \(65^{\circ}C = 65 + 273.15 = 338.15 K\).

Calculate Molecular Masses

Use the periodic table to find the molecular masses of \(O_2\) and \(UF_6\). An oxygen atom has a molecular mass of 16u, so \(O_2\) (consisting of two oxygen atoms) has a molecular mass of \(32 u = 32 \cdot 1.66053906660 \times 10^{-27} kg\). The molecular mass of \(UF_6\) (consisting of one uranium atom and six fluorine atoms) is \(238u + 19u \cdot 6 = 366u = 366 \cdot 1.66053906660 \times 10^{-27} kg\).

Calculate the Root Mean Square Speed

We use the formula for the root mean square speed, \(v_{rms} = \sqrt{{3kT} / {m}}\), where \(k\) is the Boltzmann constant (\(1.381 \times 10^{-23} J/K\)), \(T\) is the temperature in Kelvin, and \(m\) is the molecular mass in kilograms. Substitute the values into the equation for each gas to find the root mean square speed. Be sure to use the atomic mass units in kilograms.

Compare the Results

After calculating the root mean square speeds for both gases, compare the results as required in the problem statement. The gas with the higher root mean square speed has molecules that, on average, are moving faster than the other gas.

Key concepts

Molecular Mass

Understanding the molecular mass is essential when comparing properties like the root-mean-square speed of various gases. Molecular mass, also known as molecular weight, is the sum of the masses of all the atoms in a molecule computed in atomic mass units (u). For example, an oxygen molecule \(O_2\) consists of two oxygen atoms each with an atomic mass of approximately 16u. Therefore, the molecular mass of \(O_2\) is \(2 \times 16u = 32u\).

Knowing how to calculate molecular mass allows us to quantify the amount of substance and relate it to other properties such as gas speed, pressure, and temperature under various conditions. This is because the mass of the molecules directly affects how they move under a given set of circumstances, such as pressure or temperature. In the context of comparing gas speeds, heavier molecules, meaning those with higher molecular masses will typically move slower than lighter ones at the same temperature, due to having greater inertia.

Kelvin Temperature Conversion

When working with gases and their behaviors, it's crucial to use the Kelvin scale for temperature, as it is an absolute temperature scale starting from absolute zero. To convert Celsius to Kelvin, you add 273.15 to the Celsius temperature. This is essential in scientific calculations because the Kelvin scale directly relates to the kinetic energy of particles.

In this exercise, converting the given temperature from Celsius to Kelvin ensures that the root-mean-square speed is accurately calculated. The formula for root-mean-square speed involves temperature and demonstrates that as temperature increases (expressed in Kelvin), the average speed of the gas molecules also increases, assuming the gas's molecular mass remains unchanged. This Kelvin temperature conversion allows us to standardize comparisons between different gases and their respective speeds.

Boltzmann Constant

The Boltzmann constant (\(k\)) is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It essentially bridges the macroscopic and microscopic worlds, connecting temperature (a macroscopic property) with the energy of atoms and molecules (microscopic properties).

The value of the Boltzmann constant is \(1.381 \times 10^{-23} J/K\) and is used in various statistical mechanics equations, most notably in the formula for calculating the root-mean-square speed of gas molecules (\(v_{rms} = \sqrt{{3kT} / {m}}\)). This formula highlights that the average speed of particles within a gas is proportional to the square root of the temperature when expressed in Kelvin and inversely proportional to the square root of the molecular mass. The constant is indispensable in helping us understand the behavior of gases in thermodynamics.