An environmental chemist sampling industrial exhaust gases from a coal-burning plant collects a \(\mathrm{CO}_{2}-\mathrm{SO}_{2}-\mathrm{H}_{2} \mathrm{O}\) mixture in a \(21-\mathrm{L}\) steel tank until the pressure reaches \(850 .\) torr at \(45^{\circ} \mathrm{C}\) (a) How many moles of gas are collected? (b) If the \(\mathrm{SO}_{2}\) concentration in the mixture is \(7.95 \times 10^{3}\) parts per million by volume (ppmv), what is its partial pressure? [Hint: \(\mathrm{ppmv}=\left(\right.\) volume of component/volume of mixture) \(\left.\times 10^{6} .\right]\)

The Ideal Gas Law describes the relationship between pressure, volume, temperature, and the number of moles of a gas. It's expressed as \(PV = nRT\). In this formula:

• P stands for pressure, usually in atmospheres (atm).
• V is the volume, in liters (L).
• n represents the number of moles of gas.
• R is the gas constant, and it has a value of \(0.0821 \text{ L⋅atm⋅K}^{-1}\text{⋅mol}^{-1}\).
• T stands for temperature, measured in Kelvin (K).

To convert temperatures from Celsius to Kelvin, you just add 273.15. So for example, 45°C is \(45 + 273.15 = 318.15 \text{ K}\). To apply the Ideal Gas Law, simply plug in your known values. In the given problem, with \( P = 1.1184 \text{ atm}\), \( V = 21 \text{ L}\), \( R = 0.0821 \text{ L⋅atm⋅K}^{-1}\text{⋅mol}^{-1}\), and \( T = 318.15 \text{ K}\), you solve for \( n \) by rearranging the formula to \( n = \frac{PV}{RT} \). This yields \( n = \frac{(1.1184 \text{ atm})(21 \text{ L})}{(0.0821 \text{ L⋅atm⋅K}^{-1}\text{⋅mol}^{-1})(318.15 \text{ K})} \approx 0.896 \text{ moles} \).

Partial pressure is the pressure exerted by an individual gas in a mixture of gases. Dalton's Law of Partial Pressures states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of the individual gases. For example, in the given problem, the sulfur dioxide (\text{SO}_2 ) concentration is expressed in parts per million by volume (ppmv). It defines the proportion of the component gas in the total mixture. To convert ppmv to a decimal, you divide by one million. Therefore, \( 7.95 \times 10^3 \text{ ppmv} = 7.95 \times 10^{-3} \). Using the relationship between partial pressure and total pressure, partial pressure is calculated by multiplying the total pressure by the gas's fractional concentration. So, \( P_{SO_2} = P_{total} \times \text{concentration} \). Given \( P_{total} = 1.1184 \text{ atm} \) and concentration \( 7.95 \times 10^{-3} \), this becomes \( P_{SO_2} = 1.1184 \text{ atm} \times 7.95 \times 10^{-3} = 0.008886 \text{ atm} \).

Converting units is crucial in chemistry to align all measurements to compatible units. For pressure, common units are atmospheres (atm), torr, and pascals (Pa). To convert pressure from torr to atmospheres, use the conversion factor 1 atmosphere = 760 torr. For example, to convert 850 torr to atmospheres, divide by 760: \( \frac{850 \text{ torr}}{760 \text{ torr/atm}} = 1.1184 \text{ atm} \). Likewise, to revert partial pressure back to torr, multiply by 760: \( 0.008886 \text{ atm} \times 760 \text{ torr/atm} = 6.75 \text{ torr} \). Similar conversion principles apply to other units as well, ensuring consistency and accuracy in calculations. Always double-check the units in your results as doing this aids in accurate communication of scientific measurements.