Question

Given that \(1.00 \mathrm{~mol}\) of neon and \(1.00 \mathrm{~mol}\) 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 \(\mathrm{Ne} /\) density \(\mathrm{HCl}\) (c) average translational energy Ne/average translational energy HCl (d) number of Ne atoms/number of HCl molecules

Short answer

Answer: Under the given conditions, the ratios are: (a) Volume Ne/Volume HCl = 1 (b) Density Ne/Density HCl ≈ 0.553 (c) Average translational energy Ne/Average translational energy HCl = 1 (d) Number of Ne atoms/Number of HCl molecules = 1

Step-by-step solution

Volume Ne/volume HCl

Use the ideal gas law equation \({PV = nRT}\). Since we're dealing with moles, both containers have 1 mol of their respective gasses, the moles (n) will be equal. Because T (temperature) and P (pressure) are given equal and R (gas constant) is the same, V (volume) will also be equal. Therefore, the ratio of volume Ne/volume HCl is 1:1, which means the ratio is 1.

Density Ne/density HCl

To find densities, we need to find the mass of each substance. The molar mass of Ne is 20.18 g/mol and the molar mass of HCl is 1.01 (H) + 35.45 (Cl) = 36.46 g/mol. Now, use \({density(\rho) = \frac{mass}{volume}}\). As found in the previous step, the volume ratio is 1:1. So, the density ratio will be equal to the ratio of the masses: \(\frac{density_{Ne}}{density_{HCl}} = \frac{mass_{Ne}}{mass_{HCl}} = \frac{20.18}{36.46} \approx 0.553\).

Average translational energy Ne/average translational energy HCl

According to the equipartition theorem, the average translational energy is given by \({E = \frac{3}{2} kT}\), where E is the average translational energy, k is the Boltzmann constant, and T is temperature. Because temperature is the same for both, the average translational energy ratio will be the same as the degree of freedom ratio. Since both Ne and HCl are monatomic, their degree of freedom is 3. Therefore, the average translational energy ratio is \(\frac{E_{Ne}}{E_{HCl}} = \frac{3}{3} = 1\).

Number of Ne atoms/number of HCl molecules

Both containers have 1 mol of their respective substances. Since 1 mole of any substance contains Avogadro's number of particles (6.022 x 10^23), the ratio of Ne atoms to HCl molecules is \(\frac{n_{Ne}}{n_{HCl}} = \frac{1}{1} = 1\). In conclusion, (a) volume \(\mathrm{Ne} /\) volume \(\mathrm{HCl} = 1\) (b) density \(\mathrm{Ne} /\) density \(\mathrm{HCl} \approx 0.553\) (c) average translational energy Ne/average translational energy HCl = 1 (d) number of Ne atoms/number of HCl molecules = 1