Question

A sample of a smoke stack emission was collected into a 1.25-L tank at 752 mm Hg and analyzed. The analysis showed 92% CO2, 3.6% NO, 1.2% SO2, and 4.1% H2O by mass. What is the partial pressure exerted by each gas?

Short answer

Question: Calculate the partial pressures (in mm Hg) of each gas in a 1.25-L tank containing a smoke stack emission sample with mass percentages 92% CO2, 3.6% NO, 1.2% SO2, and 4.1% H2O, given a total pressure of 752 mm Hg. Answer: The partial pressures of the gases are CO2: 640 mm Hg, NO: 37 mm Hg, SO2: 5 mm Hg, and H2O: 70 mm Hg.

Step-by-step solution

Calculate the mass of each gas in the sample

Given, the sample of smoke stack emission consists of 92% CO2, 3.6% NO, 1.2% SO2, and 4.1% H2O by mass. Since the sample was collected into a 1.25-L tank, let's assume the mass of the gas sample m_tot is 100 grams (the results will be the same for any scale factor). The mass of each component can be calculated as follows: Mass of CO2 = 0.92 * m_tot Mass of NO = 0.036 * m_tot Mass of SO2 = 0.012 * m_tot Mass of H2O = 0.041 * m_tot Substitute the value of m_tot: Mass of CO2 = 92 g Mass of NO = 3.6 g Mass of SO2 = 1.2 g Mass of H2O = 4.1 g

Calculate the number of moles of each gas

Using the mass and molar mass of each gas, we can find the number of moles n for each gas component: n = mass / molar mass Molar mass of CO2 = 12.01 + 2 * 16 = 44.01 g/mol Molar mass of NO = 14.01 + 16 = 30.01 g/mol Molar mass of SO2 = 32.07 + 2 * 16 = 64.07 g/mol Molar mass of H2O = 2 * 1.01 + 16 = 18.02 g/mol n_CO2 = 92 g / 44.01 g/mol = 2.090 moles n_NO = 3.6 g / 30.01 g/mol = 0.120 moles n_SO2 = 1.2 g / 64.07 g/mol = 0.018 moles n_H2O = 4.1 g / 18.02 g/mol = 0.228 moles

Find the mole fraction of each gas

The mole fraction of each gas can be found using the formula: Mole fraction = (moles of the gas) / (total moles) Total moles (n_tot) = n_CO2 + n_NO + n_SO2 + n_H2O = 2.090 + 0.120 + 0.018 + 0.228 = 2.456 moles Mole fraction of CO2 = n_CO2 / n_tot = 2.090 / 2.456 = 0.851 Mole fraction of NO = n_NO / n_tot = 0.120 / 2.456 = 0.049 Mole fraction of SO2 = n_SO2 / n_tot = 0.018 / 2.456 = 0.007 Mole fraction of H2O = n_H2O / n_tot = 0.228 / 2.456 = 0.093

Calculate the partial pressure of each gas

Now we use the mole fraction and the total pressure (given as 752 mm Hg) to find the partial pressure of each gas component: Partial pressure = (mole fraction of the gas) * (total pressure) Partial pressure of CO2 = 0.851 * 752 mm Hg = 640 mm Hg Partial pressure of NO = 0.049 * 752 mm Hg = 37 mm Hg Partial pressure of SO2 = 0.007 * 752 mm Hg = 5 mm Hg Partial pressure of H2O = 0.093 * 752 mm Hg = 70 mm Hg So, the partial pressures of the gases in the sample are: CO2: 640 mm Hg NO: 37 mm Hg SO2: 5 mm Hg H2O: 70 mm Hg

Key concepts

Mole Fraction

Understanding mole fraction is essential when dealing with mixtures of gases or solutions. The mole fraction, often denoted by the Greek letter \( \chi \), represents the proportion of a specific component in a mixture relative to all components. To calculate the mole fraction of a gas, you divide the number of moles of that particular gas by the total number of moles of all gases in the mixture.

Using the exercise provided, if you have several gases in a container, and you know the number of moles of each, summing these values will give you the total moles present. Then, dividing each individual gas's moles by this total gives you mole fractions. For instance, if carbon dioxide's moles are 2.090 and the total moles are 2.456, the mole fraction of carbon dioxide will be \( \frac{2.090}{2.456} \), which is approximately 0.851.

Gas Laws

Gas laws are fundamental principles that describe the behavior of gases and how they respond to changes in temperature, volume, and pressure. The most relevant for partial pressure calculations is Dalton's Law of Partial Pressures, which states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases.

This is crucial when calculating the pressure each gas contributes in a mixture, like in our smoke stack emission example. As each gas in the tank behaves independently, you can find its partial pressure by multiplying its mole fraction by the total pressure. Understanding this relationship allows you to detail how gases contribute to overall pressure in their environment, leading to accurate predictions and calculations in scientific and industrial applications.

Stoichiometry

Stoichiometry is the aspect of chemistry that deals with the quantitative relationships that govern the reactants and products in chemical reactions. It's based on the conservation of mass and the concept of the mole, allowing chemists to predict the amounts of substances consumed and produced in a given reaction.

In the context of our problem, stoichiometry isn't directly involved, but understanding it can enhance your grasp of how molecules react and how mole ratios are used to calculate these quantities. Recognizing those ratios can aid in navigating complex mixtures and understanding the underlying principles of chemical reactions.

Molar Mass

Molar mass is the weight of one mole of a substance, typically expressed in grams per mole (g/mol). It's a bridge between the macroscopic world of grams and the microscopic world of molecules, allowing us to count molecules by weighing.

For the partial pressure exercise, the molar mass serves to convert the mass of each gas into moles, using the formula \( n = \frac{mass}{molar\ mass} \). For example, carbon dioxide (CO2) has a molar mass of 44.01 g/mol, so 92 grams of CO2 equate to approximately 2.090 moles. Knowing the molar mass of each gas in the smoke stack emission is integral to solving for the number of moles, which in turn is necessary for determining mole fractions and, ultimately, the partial pressures.