Question

Given that \(1.00\) mol of neon and \(1.00\) mol of hydrogen chloride gas are in separate containers at the same temperature and pressure, calculate each of the following ratios. (a) volume \(\mathrm{Ne} /\) volume \(\mathrm{HCl}\) (b) density Ne/density HCl (c) average translational energy Ne/average translational energy HCl (d) number of Ne atoms/number of HCl molecules

Short answer

Provide your answer to each of the following ratios for 1.00 mol of neon and 1.00 mol of hydrogen chloride: (a) Volume of Ne to the volume of HCl: The ratio is 1:1. (b) Density of Ne to the density of HCl: The ratio is approximately 0.55. (c) Average translational energy of Ne to the average translational energy of HCl: The ratio is 1:1. (d) Number of Ne atoms to the number of HCl molecules: The ratio is 1:1.

Step-by-step solution

(a) Volume Ne/Volume HCl

Using the ideal gas equation, since the temperature and pressure are the same for each gas, their volumes will be proportional to the number of moles. Since both gases have 1.00 mol, the ratio of their volumes will be 1:1. Therefore, the volume ratio is: \(\frac{V_{Ne}}{V_{HCl}} = 1\).

(b) Density Ne/Density HCl

The density of a gas can be expressed as \(\rho = \frac{m}{V}\), where \(\rho\) is the density, m is the mass, and V is the volume. Now, we know that the masses of 1 mole of Ne and HCl are 20.18 g/mol and 36.46 g/mol, respectively. Using the ideal gas equation, we can write the volume of the gases as \(V = \frac{nRT}{P}\). Since both gases have the same temperature, pressure, and number of moles, their volumes are the same. So, the density ratio is: \(\frac{\rho_{Ne}}{\rho_{HCl}} = \frac{m_{Ne}}{m_{HCl}}=\frac{20.18}{36.46}\approx 0.55\).

(c) Average Translational Energy Ne/Average Translational Energy HCl

The average translational energy of the gases can be represented as \(E_{trans} = \frac{3}{2}kT\), where T is the temperature and k is the Boltzmann constant. Since both gases have the same temperature, the average translational energies are the same. So, the ratio of average translational energies is: \(\frac{E_{trans, Ne}}{E_{trans, HCl}} = 1\).

(d) Number of Ne atoms/Number of HCl molecules

Both Ne and HCl have 1.00 mol, and we can use Avogadro's number (6.022 x 10^23) to calculate the number of atoms or molecules present in each. So, the ratio of the number of Ne atoms to the number of HCl molecules is: \(\frac{N_{Ne}}{N_{HCl}} = \frac{1.00\cdot (6.022 \times 10^{23})}{1.00\cdot (6.022 \times 10^{23})} = 1\).

Key concepts

Gas Volume Ratio

Understanding the gas volume ratio is crucial when comparing different gases under similar conditions. Recall the ideal gas law, which is represented by the equation: \[\begin{equation} \( PV = nRT \) \end{equation}\],where P stands for pressure, V for volume, n for moles of gas, R for the ideal gas constant, and T for temperature. This equation illustrates that, at constant pressure and temperature, the volume of a gas is directly proportional to the number of moles.

For our exercise example, both neon (Ne) and hydrogen chloride (HCl) gases were in separate containers but had the same number of moles (1.00 mol), temperature, and pressure. According to the ideal gas law, this means their volumes must also be equal, resulting in a volume ratio of 1:1. A useful tip for students: whenever gases are compared at the same temperature and pressure, expect equal volumes for equal moles, thanks to Avogadro's principle.

Gas Density Calculation

Calculating gas density, represented as the mass per unit volume (\( \rho = \frac{m}{V} \)), is another fundamental concept for understanding gaseous behavior. The mass (m) of a gas is derived from its molecular weight and the number of moles (n), as in the following:\[\begin{equation} m = n \times \text{molecular weight} \end{equation}\].When comparing the density of neon to that of hydrogen chloride in our example, we took into account their respective molecular weights of 20.18 g/mol and 36.46 g/mol. However, due to the same volume (from the ideal gas law), the ratio of densities simply comes down to the ratio of their molecular weights. The calculated density ratio of approximately 0.55 indicates that neon is less dense than hydrogen chloride under identical conditions. Remember, for any gas at a given temperature and pressure, the density will depend on its molecular weight. This concept is immensely useful in applications like predicting gas behavior when mixed or when gases are subjected to changes in environmental conditions.

Translational Energy of Gases

The translational energy of gases helps us understand the kinetic energy due to the random motion of gas particles. This energy is determined by the equation:\( E_{trans} = \frac{3}{2}kT \),where \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature. In the exercise, both gases were held at the same temperature, which implies that their average translational energies are also equal, thus giving us a ratio of 1:1.

This equivalence of energies is a demonstration of the equipartition theorem which states that energy is shared equally among all degrees of freedom. For monoatomic gases like neon and diatomic gases such as HCl, at the same temperature, each degree of freedom has the same average energy. Although these gases have different molecular structures, the translational energy only depends on temperature, reinforcing the universality of thermodynamic concepts across different types of gases.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and physics. It denotes the number of atoms, ions, or molecules in one mole of a substance and is approximately \(6.022 \times 10^{23}\).In our problem, using Avogadro's number allows us to compute the absolute quantity of Ne atoms and HCl molecules in a mole. Given that both gases have one mole each, they contain equal amounts of particles, reinforcing the notion that a mole of any substance always contains the same number of constituent particles. This number is incredibly large because atoms and molecules are exceptionally small.Avogadro's number isn't just an arbitrary value but a bridge between the microscopic world of atoms and the macroscopic quantities we can measure and observe. Remember this number when dealing with stoichiometry, interpreting gas laws, or converting between moles and particles.