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Chapter 10: Problem 63

A piece of solid carbon dioxide with a mass of \(5.50 \mathrm{~g}\) is placed in a \(10.0\) -L vessel that already contains air at 705 torr and \(24^{\circ} \mathrm{C}\). After the carbon dioxide has totally vaporized, what is the partial pressure of carbon dioxide and the total pressure in the container at \(24^{\circ} \mathrm{C}\) ?

Short Answer

Expert verified
The number of moles of CO2 is \(n = \frac{5.50 g}{44.01 g/mol} = 0.125 mol\). The temperature in Kelvin is \(T = 24掳C + 273.15 = 297.15 K\). The partial pressure of CO2 is \(P_{CO2} = \frac{(0.125 mol \times 0.0821 L\cdot atm/mol\cdot K \times 297.15 K)}{10 L} = 3.07 atm\). The initial air pressure in atm is \(P_{air} = 705 torr \times \frac{1 atm}{760 torr} = 0.927 atm\). The total pressure in the container is \(P_{total} = P_{air} + P_{CO2} = 0.927 atm + 3.07 atm = 3.997 atm\). Therefore, the partial pressure of CO2 is 3.07 atm, and the total pressure in the container is 3.997 atm.

Step by step solution

01

Calculate the number of moles of CO2

First, we need to find the number of moles of CO2. The mass (m) of CO2 is given as 5.50 g, and the molar mass (M) of CO2 is 44.01 g/mol (12.01 g/mol for C and 16.00 g/mol for each O). We can use the formula n = m/M to find the number of moles: n = m / M n = 5.50 g / 44.01 g/mol
02

Convert temperature to Kelvin

Before we use the Ideal Gas Law, we need to convert the temperature given in Celsius (24掳C) to Kelvin (K) using the formula K = C + 273.15: T = 24掳C + 273.15 T = 297.15 K
03

Determine the partial pressure of CO2

Now we can use the Ideal Gas Law (PV = nRT) to determine the partial pressure of CO2: P_CO2 脳 V = n 脳 R 脳 T where P_CO2 is the partial pressure of CO2, V is the volume of the container (10 L), n is the number of moles of CO2, R is the ideal gas constant (0.0821 L鈰卆tm/mol鈰匥), and T is the temperature in Kelvin (297.15 K). Rearrange the formula to solve for P_CO2: P_CO2 = (n 脳 R 脳 T) / V Plug in the values: P_CO2 = (5.50 g / 44.01 g/mol 脳 0.0821 L鈰卆tm/mol鈰匥 脳 297.15 K) / 10 L
04

Convert initial air pressure to atm

The initial pressure of the air in the container is given as 705 torr. To add this to the pressure of CO2, we need to convert it to atm using the conversion factor 1 atm = 760 torr: P_air = 705 torr 脳 (1 atm / 760 torr)
05

Calculate total pressure in the container

Now we can sum the initial air pressure (in atm) and the partial pressure of CO2 (in atm) to find the total pressure in the container: P_total = P_air + P_CO2 Plug in the values from previous steps to calculate P_total.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Law
Understanding the Ideal Gas Law is key to solving problems involving gases in containers. It is represented by the equation \(PV = nRT\), where \(P\) stands for the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the number of moles of gas, \(R\) is the ideal gas constant (0.0821 L鈰卆tm/mol鈰匥 is commonly used), and \(T\) is the absolute temperature in Kelvin. This law combines several gas laws, including Boyle's, Charles', and Avogadro's, into a single formula.

In the context of the exercise, the Ideal Gas Law allows us to calculate the partial pressure of carbon dioxide in a container. The partial pressure is the pressure the gas would exert if it were the only gas in the container. Since the vessel contains air, the carbon dioxide will have a partial pressure that contributes to the total pressure of the mixture. It's crucial to use Kelvin for the temperature and to make sure the volume is in liters and the pressure in atmospheres to match the units of the gas constant when using this formula.
Molar Mass Calculation
The molar mass of a substance is the mass of one mole of that substance, and is expressed in grams per mole (\(g/mol\)). To calculate it for a compound, you must sum the molar masses of the individual elements that make up the compound, taking into account their respective quantities as seen in the chemical formula.

For example, to find the molar mass of carbon dioxide (\(CO_2\)), you must account for one carbon atom and two oxygen atoms. Carbon has a molar mass of about 12.01 \(g/mol\), and oxygen has a molar mass of about 16.00 \(g/mol\). Therefore, the molar mass of \(CO_2\) is \(12.01 g/mol + 2(16.00 g/mol) = 44.01 g/mol\). The exercise asked us to use this molar mass to calculate the number of moles of carbon dioxide from its given mass of 5.50 grams. Understanding molar mass is essential for stoichiometry in chemistry, which involves calculations based on the mass of reactants and products in a chemical reaction.
Converting Temperature to Kelvin
Kelvin is the SI unit for temperature and is one of the fundamental units in physics. The Kelvin scale is an absolute temperature scale starting at absolute zero, which is the theoretical point where particles have minimum thermal motion. To convert Celsius to Kelvin, which is necessary in gas law calculations including the Ideal Gas Law, you use the formula \(K = ^\circ C + 273.15\).

In our exercise, the temperature given is \(24^\circ C\), so we converted that to Kelvin by adding 273.15, resulting in \(297.15 K\). It's critical that this conversion is done because the Ideal Gas Law requires the absolute temperature in Kelvin for accurate calculations. This ensures that all the components in the equation are consistent, allowing us to accurately assess how temperature influences the behavior of gases.

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Most popular questions from this chapter

A mixture of gases contains \(0.75 \mathrm{~mol} \mathrm{~N}_{2}, 0.30 \mathrm{~mol} \mathrm{O}_{2}\) and \(0.15 \mathrm{~mol} \mathrm{CO}_{2}\). If the total pressure of the mixture is \(1.56 \mathrm{~atm}\), what is the partial pressure of each component?

Carbon dioxide, which is recognized as the major contributor to global warming as a "greenhouse gas," is formed when fossil fuels are combusted, as in electrical power plants fueled by coal, oil, or natural gas. One potential way to reduce the amount of \(\mathrm{CO}_{2}\) added to the atmosphere is to store it as a compressed gas in underground formations. Consider a 1000 -megawatt coalfired power plant that produces about \(6 \times 10^{6}\) tons of \(\mathrm{CO}_{2}\) per year. (a) Assuming ideal gas behavior, \(1.00 \mathrm{~atm}\), and \(27{ }^{\circ} \mathrm{C}\), calculate the volume of \(\mathrm{CO}_{2}\) produced by this power plant. (b) If the \(\mathrm{CO}_{2}\) is stored underground as a liquid at \(10^{\circ} \mathrm{C}\) and \(120 \mathrm{~atm}\) and a density of \(1.2 \mathrm{~g} / \mathrm{cm}^{3}\), what volume does it possess? (c) If it is stored underground as a gas at \(36{ }^{\circ} \mathrm{C}\) and \(90 \mathrm{~atm}\), what volume does it occupy?

A gas forms when elemental sulfur is heated carefully with AgF. The initial product boils at \(15^{\circ} \mathrm{C}\). Experiments on several samples yielded a gas density of \(0.803 \pm 0.010 \mathrm{~g} / \mathrm{L}\) for the gas at \(150 \mathrm{~mm}\) pressure and \(32{ }^{\circ} \mathrm{C}\). When the gas reacts with water, all the fluorine is converted to aqueous HF. Other products are elemental sulfur, \(S_{8}\), and other sulfur-containing compounds. A 480 -mL sample of the dry gas at \(126 \mathrm{~mm}\) pressure and \(28^{\circ} \mathrm{C}\), when reacted with \(80 \mathrm{~mL}\) of water, yielded a \(0.081 \mathrm{M}\) solution of HF. The initial gaseous product undergoes a transformation over a period of time to a second compound with the same empirical and molecular formula, which boils at \(-10^{\circ} \mathrm{C}\). (a) Determine the empirical and molecular formulas of the first compound formed. (b) Draw at least two reasonable Lewis structures that represent the initial compound and the one into which it is transformed over time. (c) Describe the likely geometries of these compounds, and estimate the single bond distances, given that the \(\mathrm{S}-\mathrm{S}\) bond distance in \(\mathrm{S}_{8}\) is \(2.04 \mathrm{~A}\) and the \(\mathrm{F}-\mathrm{F}\) distance in \(\mathrm{F}_{2}\) is \(1.43 \mathrm{~A}\).

Consider a mixture of two gases, \(A\) and \(B\), confined in a closed vessel. A quantity of a third gas, \(\mathrm{C}\), is added to the same vessel at the same temperature. How does the addition of gas \(\mathrm{C}\) affect the following: (a) the partial pressure of gas \(A,(b)\) the total pressure in the vessel, (c) the mole fraction of gas B?

(a) How is the law of combining volumes explained by Avogadro's hypothesis? (b) Consider a 1.0-L flask containing neon gas and a 1.5-L flask containing xenon gas. Both gases are at the same pressure and temperature. According to Avogadro's law, what can be said about the ratio of the number of atoms in the two flasks?
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