Problem 84 - Chemistry The Molecular Nature of Matter

Question

A special gas mixture, BAR 97 High without \(\mathrm{NO},\) is used in engine emission testing and contains \(16.3 \mathrm{~atm} \mathrm{CO}_{2}\), 8270 torr \(\mathrm{CO},\) and \(331 \mathrm{~mm} \mathrm{Hg}\) propane with the 108 atm nitrogen. What was the total pressure in the gas cylinder?

Step-by-Step Solution

Answer

The total pressure in the gas cylinder is 135.62 atm.

Step 1: Convert all pressures to the same unit

Before adding up the partial pressures, we need to convert all pressures to the same unit. We'll convert torr and mm Hg to atm because the final answer is most commonly expressed in atm for gas mixtures. The conversion factor is 1 atm = 760 mm Hg or torr. Thus, for CO we have: \(8270 \text{ torr} \times \frac{1 \text{ atm}}{760 \text{ torr}} = 10.88 \text{ atm}\) and for propane, \(331 \text{ mm Hg} \times \frac{1 \text{ atm}}{760 \text{ mm Hg}} = 0.44 \text{ atm}\).

Step 2: Sum up all partial pressures

Add the pressures of CO2, CO, propane, and nitrogen to find the total pressure. The individual pressures in atm are: CO2 = 16.3 atm, CO = 10.88 atm, propane = 0.44 atm, and nitrogen = 108 atm. The total pressure is the sum of these values.

Step 3: Calculate the total pressure

The total pressure (\(P_{\text{total}}\)) is given by \(P_{\text{total}} = P_{\text{CO}_2} + P_{\text{CO}} + P_{\text{propane}} + P_{\text{N}_2} = 16.3 \text{ atm} + 10.88 \text{ atm} + 0.44 \text{ atm} + 108 \text{ atm} = 135.62 \text{ atm}\).

Key Concepts

Partial Pressure

The concept of partial pressure is fundamental when dealing with gas mixtures. It refers to the pressure that a single component of a gas mixture would exert if it occupied the entire volume of the mixture at the same temperature. This concept is particularly important because it allows us to treat each gas in a mixture independently according to Dalton's Law of Partial Pressures, which states that the total pressure of a mixture is equal to the sum of the partial pressures of each component gas.

Understanding partial pressure is essential for predicting gas behavior in a range of scientific and real-world applications such as scuba diving, where knowing the partial pressure of oxygen can prevent oxygen toxicity, and in medicine where the oxygen content of blood must be regulated. For students, it's valuable to associate this particularly with gas laws and stoichiometry in chemical reactions.

Pressure Unit Conversion

Pressure unit conversion is a common necessity in chemistry and physics, as various units are used depending on the context. The most common pressure units include atmosphere (atm), torr, millimeter of mercury (mm Hg), pascal (Pa), and bar. Conversion between these units requires understanding the equivalence values such as 1 atm equals 760 torr or 760 mm Hg.

When solving problems with multiple pressure readings, as in our textbook exercise, converting all measurements to a single unit simplifies arithmetic and prevents errors. Practical applications abound; for example, meteorologists convert pressure units for weather forecasts, and engineers do the same when designing pressure-dependent systems. An important tip for students is always to keep track of units throughout calculations to ensure accurate final results.

Ideal Gas Law

The concept of partial pressure is fundamental when dealing with gas mixtures. It refers to the pressure that a single component of a gas mixture would exert if it occupied the entire volume of the mixture at the same temperature. This concept is particularly important because it allows us to treat each gas in a mixture independently according to Dalton's Law of Partial Pressures, which states that the total pressure of a mixture is equal to the sum of the partial pressures of each component gas.

Understanding partial pressure is essential for predicting gas behavior in a range of scientific and real-world applications such as scuba diving, where knowing the partial pressure of oxygen can prevent oxygen toxicity, and in medicine where the oxygen content of blood must be regulated. For students, it's valuable to associate this particularly with gas laws and stoichiometry in chemical reactions.