Problem 50 - Chemistry The Central Science

Question

Rank the following gases from least dense to most dense at 1.00 atm and \(298 \mathrm{~K}: \mathrm{SO}_{2}, \mathrm{HBr}, \mathrm{CO}_{2} .\) Explain.

Step-by-Step Solution

Answer

The densities of the gases at 1.00 atm and 298 K are as follows: CO2 has a density of 1.82 g/L, SO2 has a density of 2.63 g/L, and HBr has a density of 3.29 g/L. Therefore, the gases can be ranked from least dense to most dense in the order: CO2, SO2, HBr.

Step 1: Write down the ideal gas law

The ideal gas law is given by the formula: \(PV = nRT\) where P is the pressure of the gas, V is its volume, n is the number of moles of the gas, R is the ideal gas constant, and T is the temperature of the gas.

Step 2: Derive the formula for density

The density of the gas (\(\rho\)) can be calculated using the formula: \(\rho = \frac{m}{V}\) where m is the mass of the gas and V is its volume. We can rewrite the mass in terms of the number of moles (n) and molar mass (M) as follows: \(m = nM\) We can substitute this expression into the density formula: \(\rho = \frac{nM}{V}\) Now, we can use the ideal gas law to relate V and n: \(PV = nRT\) \(\frac{V}{n} = \frac{RT}{P}\) \(\frac{1}{V} = \frac{P}{RT}\) Substitute this expression for \(\frac{1}{V}\) into the density formula: \(\rho = \frac{nM}{\frac{RT}{P}}\) \(\rho = \frac{MP}{RT}\)

Step 3: Calculate the densities of each gas

We are given that the pressure is 1.00 atm and the temperature is 298 K. We can now calculate the molar mass of each gas and then use the density formula to determine their densities. Molar mass of SO2: M_SO2 = 32.07 (S) + 2 * 16.00 (O) = 64.07 g/mol Molar mass of HBr: M_HBr = 1.01 (H) + 79.90 (Br) = 80.91 g/mol Molar mass of CO2: M_CO2 = 12.01 (C) + 2 * 16.00 (O) = 44.01 g/mol Density of SO2 (\(\rho_{SO2}\)): \(\rho_{SO2} = \frac{(64.07\times1.00)}{(0.0821\times298)} = 2.63 \frac{g}{L}\) Density of HBr (\(\rho_{HBr}\)): \(\rho_{HBr} = \frac{(80.91\times1.00)}{(0.0821\times298)} = 3.29 \frac{g}{L}\) Density of CO2 (\(\rho_{CO2}\)): \(\rho_{CO2} = \frac{(44.01\times1.00)}{(0.0821\times298)} = 1.82 \frac{g}{L}\)

Step 4: Rank the densities

From the calculated densities, we can rank the gases from least dense to most dense as follows: CO2 (1.82 g/L) < SO2 (2.63 g/L) < HBr (3.29 g/L)

Key Concepts

Ideal Gas Law

The foundation of understanding gas density lies in the ideal gas law. This law provides a simple equation that relates several important physical properties of gases: pressure (P), volume (V), number of moles (n), the ideal gas constant (R), and temperature (T). In the context of the ideal gas law, the equation is written as: \[ PV = nRT \] This equation is immensely helpful in predicting how a real gas might behave under certain conditions, although it is an idealization. It assumes that gases are made up of non-interacting point particles that collide elastically. By manipulating this equation, for instance, rearranging it to find other variables or derive expressions for physical quantities like density, we get a better grip on how gases work. Understanding and applying the ideal gas law is crucial for solving problems related to gases.

Molar Mass

Molar mass is a key concept when dealing with gases, especially in the context of calculating their density. It represents the mass of a given substance (a mole of molecules) and is typically expressed in grams per mole (g/mol). You can think of molar mass as a 'weight' of one mole's worth of a particular element or compound.

Calculating Molar Mass

To determine the molar mass of a compound, you sum up the atomic masses of all the atoms present in a single molecule of that compound. For instance: - Molar mass of sulfur dioxide (\( SO_2 \)): Sulfur (32.07 g/mol) plus 2 oxygens (16.00 g/mol each) results in 64.07 g/mol- Similarly, calculate for other gases using their molecular composition. Understanding molar mass not only helps in density calculations but also in converting between moles and grams, a fundamental skill in chemistry.

Gas Ranking

Gas ranking by density involves comparing gases to determine which is heavier or lighter in a given condition like constant pressure and temperature. Here, density is determined by a formula derived from the ideal gas law: \[ \rho = \frac{MP}{RT} \] where \( \rho \) is density, M is molar mass, P is pressure, R is the ideal gas constant, and T is temperature. Since both pressure and temperature are constants in our problem, differences in density are mainly due to differences in molar mass.

Ranking Gases

To rank gases as least dense to most dense: - Calculate the density of each gas using its molar mass.- A higher molar mass implies a higher density when conditions are constant. For the given gases, CO2 is the least dense, followed by SO2, and HBr being the most dense. This sequence helps in understanding gases' behavior and potential uses or impacts in real-world scenarios.