A rigid, sealed 12.00 \(\mathrm{L}\) container is filled with 10.00 \(\mathrm{g}\) each of three different gases: \(\mathrm{CO}_{2}, \mathrm{NO},\) and \(\mathrm{NH}_{3}\) . The temperature of the gases is held constant \(35.0^{\circ} \mathrm{C} .\) Assume ideal behavior for all gases. (a) (i) What is the mole fraction of each gas? (ii) What is the partial pressure of each gas? (b) Out of the three gases, molecules of which gas will have the highest velocity? Why? (c) Name one circumstance in which the gases might deviate from ideal behavior, and clearly explain the reason for the deviation.

Moles of a gas can be calculated by using the formula: \(\frac{composition (grams)}{molecular weight}\). Using this, we find: Moles of \(\mathrm{CO}_{2} = \frac{10.00 \, g}{44.01 \, g/mol} = 0.227 \, mol\), \(\mathrm{NO} = \frac{10.00 \, g}{30.01 \, g/mol} = 0.333 \, mol \), and \(\mathrm{NH}_{3} = \frac{10.00 \, g}{17.03 \, g/mol} = 0.587 \, mol\). Total moles is the sum of moles of \(\mathrm{CO}_{2}, \mathrm{NO},\) and \(\mathrm{NH}_{3}\), i.e., \(0.227+0.333+0.587 = 1.147 \, mol\). Mole fraction of a gas is the ratio of the number of moles of that gas to the total number of moles of all gases. Using this formula, we find mole fractions of \(\mathrm{CO}_{2}, \mathrm{NO},\) and \(\mathrm{NH}_{3}\) respectively: \(\frac{0.227}{1.147} \approx 0.198\), \(\frac{0.333}{1.147} \approx 0.290\), \(\frac{0.587}{1.147} \approx 0.512 \).

The total pressure \(P_{total}\) of the gases can be found using the ideal gas law \(\left(PV=nRT\right)\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is number of moles, \(R\) is the gas constant (0.0821 L atm mol⁻¹K⁻¹), and \(T\) is the temperature in Kelvin. The room temperature in Kelvin can be found by \(T(K) = T(°C) + 273.15 = 35.0 + 273.15 = 308.15K\). Using these values in the ideal gas law gives: \(P_{total}=\frac{nRT}{V}=\frac{(1.147\,mol)(0.0821 \, L \, atm \, mol⁻¹K⁻¹)(308.15K)}{12.00L}=2.494 \, atm\). The partial pressure of a gas is given by \(P_{gas}=x_{gas}*P_{total}\), where \(x_{gas}\) is the mole fraction of the gas. The partial pressures of \(\mathrm{CO}_{2}, \mathrm{NO},\) and \(\mathrm{NH}_{3}\) are: \(\approx 0.494 \, atm\), \(\approx 0.725 \, atm\), and \(\approx 1.277 \, atm\) respectively.

Using Graham's law of diffusion, the rate of diffusion (or the average velocity of a gas) is inversely proportional to the square root of its molar mass. Therefore, the gas with the lowest molecular weight will have the fastest average speed. Here, \(\mathrm{NH}_{3}\) has the lowest molecular weight (17.03 g/mol) and hence, it will have the highest velocity.

The gases might deviate from ideal behavior under conditions of high pressure and low temperature. This is because as the pressure increases, the volume becomes very small, causing the molecules to be in close proximity to each other. Under such conditions, the interaction between molecules becomes significant which is contrary to the assumption of ideal gases that there are no forces of attraction or repulsion between the gas molecules. Similarly, at low temperatures, the kinetic energy of the molecules decreases and forces of attraction among molecules become more prominent, again, leading to deviation from ideal behavior.