Problem 47 - Chemistry The Central Science In Si Units

Question

Rank the following gases from least dense to most dense at \(101.33 \mathrm{kPa}\) and \(298 \mathrm{~K}: \mathrm{O}_{2}, \mathrm{Ar}, \mathrm{NH}_{3}, \mathrm{HCl}\)

Step-by-Step Solution

Answer

The gases ranked from least dense to most dense are: NH₃, O₂, HCl, Ar.

Step 1: Understand the concept of density

Gas density at a particular temperature and pressure can be calculated using the equation \(\rho = \frac{PM}{RT}\), where \(\rho\) is the density, \(P\) is the pressure, \(M\) is the molar mass, \(R\) is the universal gas constant, and \(T\) is the temperature. Density is directly proportional to molar mass when temperature and pressure are constant.

Step 2: Determine the molar mass of each gas

Calculate the molar mass of each gas: Hydrogen Chloride (HCl) has a molar mass of approximately 36.46 g/mol, Argon (Ar) has a molar mass of 39.95 g/mol, Oxygen (O₂) has a molar mass of 32.00 g/mol, and Ammonia (NH₃) has a molar mass of approximately 17.03 g/mol.

Step 3: Rank the gases by molar mass

Rank the gases from least to most molar mass. Using the calculated molar masses: NH₃ (17.03 g/mol), O₂ (32.00 g/mol), HCl (36.46 g/mol), and Ar (39.95 g/mol).

Step 4: Conclude the ranking by density

Since density is directly proportional to molar mass in this context, the ranking of gases from least dense to most dense is the same as the ranking by molar mass: NH₃, O₂, HCl, Ar.

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry that refers to the mass of one mole of a substance. It is expressed in grams per mole (g/mol) and serves as a bridge between the atomic scale and weights we can measure in the lab. To calculate molar mass, you need to sum up the atomic masses of all the atoms present in a molecule. For example, to determine the molar mass of water (\(H_2O\)), add up the atomic masses of two hydrogen atoms (approximately 1 g/mol each) and one oxygen atom (approximately 16 g/mol), resulting in roughly 18 g/mol. This same approach applies to more complex molecules like \(NH_3\) or any other compounds.

• Knowing the molar mass allows you to relate mass measures of a substance to its chemical proportions.
• Molar mass is essential in stoichiometry calculations, helping predict the amounts of products and reactants involved in chemical reactions.

In the context of gases, like in our exercise, molar mass is crucial for estimating gas density under constant temperature and pressure conditions. This makes it easier for scientists to understand and predict how gases will behave in different situations.

Ideal Gas Law

The Ideal Gas Law is a vital equation in chemistry that describes the behavior of an ideal gas. It is expressed in the formula:\[ PV = nRT \] where:- \( P \) is the pressure- \( V \) is the volume- \( n \) is the number of moles- \( R \) is the universal gas constant- \( T \) is the temperature in Kelvin.This equation shows the relationship between pressure, volume, temperature, and number of moles of a gas. By rearranging this equation, you can solve for other properties. For example, to find the density (\(\rho\)) of a gas, you can use the derived equation:\[ \rho = \frac{PM}{RT} \] Here, \( M \) denotes the molar mass, and through this formula, density becomes a function of pressure, molar mass, temperature, and the universal gas constant.

• The Ideal Gas Law is a foundational tool for predicting gas behavior under ideal conditions, and it is still a good approximation for many gases.
• Understanding this law helps in correlating foundational concepts like pressure, volume, and temperature with physical properties like density.

This law is immensely useful in ranking gases by density when temperature and pressure are consistent, by analyzing their molar masses.

Temperature and Pressure

Temperature and pressure are two critical factors influencing the behavior of gases. These variables directly affect gas volume, energy, and density. - **Temperature**: It refers to the measure of the average kinetic energy of particles in a substance. For gases, as temperature increases, the kinetic energy of gas molecules also increases, causing them to move faster. - **Pressure**: Pressure measures the force exerted by gas molecules colliding against the surfaces of their containers. Higher temperatures often lead to increased pressure if the volume is constant, as particles collide more frequently and forcefully. In gas density calculations, temperature and pressure are vital because they directly impact the state and behavior of a gas. Consider, for instance: At higher temperatures, a gas expands because of increased particle motion, often resulting in a lower density. Likewise, if the pressure is increased, the gas molecules are compressed, which increases the density.

• To understand gas behavior using these parameters, remember that under standard conditions, gases exhibit predictable behaviors that align with the Ideal Gas Law.
• Maintaining constant temperature and pressure allows scientists to study other properties, like molar mass, to determine gas density accurately.

By holding temperature and pressure consistent, you can precisely evaluate a gas's density based on factors like its molar mass.