Question

A 5.0-L flask contains \(0.60 \mathrm{~g} \mathrm{O}_{2}\) at a temperature of \(22^{\circ} \mathrm{C}\). What is the pressure (in atm) inside the flask?

Short answer

The pressure inside the 5.0-L flask containing 0.60 g of O2 at a temperature of 22°C is approximately \(0.726 \mathrm{~atm}\).

Step-by-step solution

Convert the temperature from Celsius to Kelvin

To convert the temperature from Celsius to Kelvin, we need to add 273.15 to the given temperature in Celsius. So the Kelvin temperature (T) is: \(T = 22 + 273.15 = 295.15 \mathrm{~K} \)

Convert the mass of oxygen into moles

We are given the mass of O2 in grams, but we need the amount in moles to use the Ideal Gas Law. To convert the mass to moles, we will use the molar mass of O2. The molar mass of O2 is 32 g/mol (since there are 2 oxygen atoms, each with a mass of roughly 16 g/mol). So, we can calculate the number of moles (n) by dividing the mass by the molar mass: \( n = \dfrac{Mass_{O_2}}{Molar~mass_{O_2}} \) \( n = \dfrac{0.60 \mathrm{~g}}{32 \mathrm{~g/mol}} = 0.01875 \mathrm{~mol}\)

Use the Ideal Gas Law to find the pressure

Now, we can use the Ideal Gas Law equation to find the pressure, with the given volume and calculated values for moles and temperature: Ideal Gas Law: \(PV = nRT \) where P is the pressure we want to find, V is the volume (5.0 L), n is the number of moles (0.01875 mol), R is the ideal gas constant (0.0821 L·atm/mol·K, using atm unit for pressure), and T is the temperature in Kelvin (295.15 K). Rearrange the equation to find the pressure: \( P = \dfrac{nRT}{V} \) Plug in the values: \( P = \dfrac{(0.01875 \mathrm{~mol})(0.0821 \mathrm{~L~atm/mol~K})(295.15 \mathrm{~K})}{5.0 \mathrm{~L}} \) Calculate the pressure: \(P \approx 0.726 \mathrm{~atm} \) So, the pressure inside the flask is approximately 0.726 atm.

Key concepts

Calculating Pressure

Understanding how to calculate pressure is fundamental in chemistry, especially when working with gases. Pressure can be thought of as the force exerted by gas particles as they collide with the walls of their container. This force per unit area is what we know as pressure. In the provided exercise, the Ideal Gas Law is used to calculate the pressure of oxygen gas inside a flask.

To calculate the pressure using the Ideal Gas Law equation, we need four important values: the number of moles of gas, the temperature in Kelvin, the volume of the gas, and the ideal gas constant. Once we have these, we can rearrange the Ideal Gas Law, which is expressed as \(PV = nRT\), to solve for pressure (P). The significance of the Ideal Gas Law lies in its ability to connect these variables in a single equation that applies to ideal gases, or real gases under certain conditions.

The process includes measuring or estimating the number of moles of gas, the volume of the gas's container, and the gas's temperature, which we then adjust using the appropriate gas constant for the units we're working with. By plugging these values into the equation and solving for P, we can find the pressure of the gas.

Conversion of Temperature to Kelvin

Temperature plays a vital role in gas behavior, and for gas law calculations, it is measured in Kelvin—the base unit of temperature in the International System of Units (SI). To convert Celsius to Kelvin, we add 273.15 to the Celsius temperature. This step is crucial because it aligns the temperature scale with the absolute zero point, where theoretically there is no kinetic energy in the particles of the substance.

For instance, in our exercise, the temperature given is \(22^{\textdegree} \text{C}\), which we convert to Kelvin by adding 273.15, resulting in \(295.15 \text{K}\). Using Kelvin allows us to avoid negative temperature values, which is important because the laws of thermodynamics imply that there's no such thing as a negative absolute temperature. This ensures that the mathematical relationships in our gas law calculations remain consistent and meaningful.

Mole Concept in Chemistry

The mole concept is fundamental to stoichiometry, which is the study of the quantitative aspects of chemical reactions and substances. In chemistry, the mole is used as a unit to express amounts of a chemical substance. It allows chemists to relate the mass of substances to the number of particles, such as molecules or atoms, since directly counting the number of particles is not feasible.

One mole is defined as the amount of substance that contains as many particles as there are atoms in 12 grams of pure carbon-12. This number is Avogadro's number (approximately \(6.022 \times 10^{23}\) entities). In our exercise, we convert the mass of oxygen to moles to use it in the Ideal Gas Law. We do this because the law deals with the number of particles indirectly through the mole concept, thus simplifying the calculation.

Molar Mass of Gases

Molar mass is another core concept in chemistry which refers to the mass of one mole of a substance, typically measured in grams per mole (g/mol). For gases, the molar mass tells us how much one mole of that gas weighs. It is a bridge that allows chemists to convert from mass, which is measurable, to moles, which relates directly to the number of particles.

To calculate the molar mass of a diatomic gas like oxygen (\(O_2\)), we add up the molar masses of the individual atoms. Oxygen has a molar mass of about 16 g/mol, so \(O_2\) has a molar mass of approximately 32 g/mol. Knowing the molar mass is crucial when we're converting between the mass of a gas and the number of moles for gas law calculations, as demonstrated in the exercise, which is a common task in various chemistry applications.