Problem 32
Question
A gaseous mixture of three gases \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) has a pressure of \(10 \mathrm{~atm}\). The total number of moles of all the gases is \(10 .\) If the partial pressures of \(\mathrm{A}\) and \(\mathrm{B}\) are \(3.0\) and \(1.0\) atm, respectively, and if \(C\) has molecular mass of \(2.0\), what is the mass of \(\mathrm{C}\), in \(\mathrm{g}\), present in the mixture? (a) 6 (b) 8 (c) 12 (d) 3
Step-by-Step Solution
Verified Answer
The mass of Gas C in the mixture is 8 g.
1Step 1: Determine the partial pressure of Gas C
The total pressure of the gas mixture is the sum of the partial pressures of gases A, B, and C. Given the total pressure is 10 atm and the partial pressures of A and B are 3.0 atm and 1.0 atm respectively, subtract the sum of the pressures of A and B from the total pressure to find the pressure of C.
2Step 2: Use Dalton's Law to find the moles of Gas C
Dalton's Law states that the partial pressure of a gas is proportional to its mole fraction in a mixture. With the total moles being 10 and having found the pressure of C, calculate the mole fraction and then find the moles of Gas C.
3Step 3: Calculate the mass of Gas C
With the molecular mass of Gas C given as 2.0 g/mol and the number of moles calculated in the previous step, multiply them to find the mass of Gas C in grams.
Key Concepts
Dalton's Law of Partial PressuresMole FractionMolecular Mass
Dalton's Law of Partial Pressures
Dalton's Law of Partial Pressures is a principle in chemistry that helps us understand the behavior of gas mixtures. It states that in a mixture of non-reacting gases, the total pressure exerted by the combination is equal to the sum of the partial pressures of each individual gas. A partial pressure is the pressure that a single gas in a mixture would exert if it occupied the entire volume by itself while at the same temperature.
Think of it like a team sport where the total score is the sum of points scored by each player. In a gas mixture, each gas 'contributes' its own 'score'—its partial pressure—to the total pressure. This law makes it easy to calculate how much pressure one gas is adding to the mix, just like calculating an individual player's contribution to the total score.
To apply this law for calculation, you identify the total pressure and subtract the known partial pressures of the other gases, as seen in the example problem. This allows you to determine the partial pressure of the unknown gas. Understanding this concept is essential for solving a variety of problems in gaseous systems, including those in environmental science, engineering, and of course, chemistry.
Think of it like a team sport where the total score is the sum of points scored by each player. In a gas mixture, each gas 'contributes' its own 'score'—its partial pressure—to the total pressure. This law makes it easy to calculate how much pressure one gas is adding to the mix, just like calculating an individual player's contribution to the total score.
To apply this law for calculation, you identify the total pressure and subtract the known partial pressures of the other gases, as seen in the example problem. This allows you to determine the partial pressure of the unknown gas. Understanding this concept is essential for solving a variety of problems in gaseous systems, including those in environmental science, engineering, and of course, chemistry.
Mole Fraction
The mole fraction is another key concept in chemistry, especially when dealing with mixtures of substances. It gives us a way to express the composition of a mixture by showing the proportion of the total moles represented by each component. You can calculate the mole fraction of a gas by dividing the number of moles of that gas by the total number of moles in the entire mixture.
Imagine having a bag of mixed candies where each candy type represents a different gas. The mole fraction would be the count of one type of candy divided by the total count of all the candies in the bag. This fraction is dimensionless and always less than or equal to one. When dealing with gas mixtures, the mole fraction becomes particularly useful in conjunction with Dalton's Law of Partial Pressures. With it, you can relate the partial pressure of a gas to its mole fraction in the mixture, allowing for detailed calculations regarding the composition and properties of the gas mixture.
Imagine having a bag of mixed candies where each candy type represents a different gas. The mole fraction would be the count of one type of candy divided by the total count of all the candies in the bag. This fraction is dimensionless and always less than or equal to one. When dealing with gas mixtures, the mole fraction becomes particularly useful in conjunction with Dalton's Law of Partial Pressures. With it, you can relate the partial pressure of a gas to its mole fraction in the mixture, allowing for detailed calculations regarding the composition and properties of the gas mixture.
Molecular Mass
Molecular mass, sometimes referred to as molecular weight, is the sum of the atomic masses of the atoms in a molecule. It's measured in atomic mass units (amu), or grams per mole (g/mol) when dealing with larger quantities.
When you're cooking and you need to measure ingredients, you might use cups or tablespoons. In chemistry, to 'measure' molecules, we use their molecular mass. It's a way to convey how 'heavy' a molecule is compared to a standard, which is the carbon-12 isotope. Knowing the molecular mass is essential for converting between moles and grams of a substance, as seen in our example problem.
Once the number of moles of a gas in a mixture is known, the molecular mass allows us to convert these moles into mass, which can be weighed. This is crucial for tasks ranging from calculating medication dosages to designing chemical reactions in industrial processes.
When you're cooking and you need to measure ingredients, you might use cups or tablespoons. In chemistry, to 'measure' molecules, we use their molecular mass. It's a way to convey how 'heavy' a molecule is compared to a standard, which is the carbon-12 isotope. Knowing the molecular mass is essential for converting between moles and grams of a substance, as seen in our example problem.
Once the number of moles of a gas in a mixture is known, the molecular mass allows us to convert these moles into mass, which can be weighed. This is crucial for tasks ranging from calculating medication dosages to designing chemical reactions in industrial processes.
Other exercises in this chapter
Problem 28
An amount of 1 mole of a gas is changed from its initial state \((20 \mathrm{~L}, 2 \mathrm{~atm})\) to final state (4L, \(10 \mathrm{~atm}\) ), respectively. I
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At which of the following four conditions, the density of an ideal gas will be maximum? (a) \(273 \mathrm{~K}\) and \(1 \mathrm{~atm}\) (b) \(273 \mathrm{~K}\)
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