Problem 84 - Chemistry: The Central Science

Question

(a) Place the following gases in order of increasing average molecular speed at \(300 \mathrm{~K}: \mathrm{CO}, \mathrm{SF}_{6}, \mathrm{H}_{2} \mathrm{~S}, \mathrm{Cl}_{2}, \mathrm{HBr}\). (b) Calculate the rms speeds of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) molecules at \(300 \mathrm{~K}\). (c) Calculate the most probable speeds of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) molecules at \(300 \mathrm{~K}\).

Step-by-Step Solution

Answer

(a) The order of increasing average molecular speed is \(SF_{6} < HBr < Cl_{2} < H_{2}S < CO\). (b) The rms speeds are \(v_{\mathrm{rms} (\mathrm{CO})} = 430.50 \ \mathrm{m/s}\) and \(v_{\mathrm{rms} (\mathrm{Cl_{2})} = 301.59 \ \mathrm{m/s}\). (c) The most probable speeds are \(v_{\mathrm{p} (\mathrm{CO})} = 384.91 \ \mathrm{m/s}\) and \(v_{\mathrm{p} (\mathrm{Cl_{2})} = 269.73 \ \mathrm{m/s}\).

Step 1: Calculate Molar Masses

Firstly, we need to calculate the molar masses of each gas. Molar mass is the mass of one mole of a substance and it is the sum of the atomic masses of the atoms in a molecule. - CO: \(\mathrm{C} = 12.01 \ \mathrm{g/mol}, \ \mathrm{O} = 16.00 \ \mathrm{g/mol} \\ M_{\mathrm{CO}} = 12.01+16.00 = 28.01 \mathrm{~g/mol}\) - SF6: \(\mathrm{S} = 32.07 \ \mathrm{g/mol}, \ \mathrm{F} = 19.00\ \mathrm{g/mol} \\ M_{\mathrm{SF_{6}}} = 32.07 + 6\cdot19.00 = 146.07\ \mathrm{g/mol}\) - H2S: \(\mathrm{H}=1.01 \ \mathrm{g/mol}, \ \mathrm{S}=32.07 \ \mathrm{g/mol} \\ M_{\mathrm{H_{2} S}} = 2(1.01) + 32.07 = 34.09\ \mathrm{g/mol}\) - Cl2: \(\mathrm{Cl} = 35.45 \ \mathrm{g/mol} \\ M_{\mathrm{Cl_{2}}} = 2(35.45) = 70.90\ \mathrm{g/mol}\) - HBr: \(\mathrm{H} = 1.01\ \mathrm{g/mol}, \ \mathrm{Br} = 79.90\ \mathrm{g/mol} \\ M_{\mathrm{HBr}} = 1.01 + 79.90 = 80.91\ \mathrm{g/mol}\)

Step 2: Order by Average Molecular Speed

To find the order of increasing average molecular speed, we can use the formula for average molecular speed (for an ideal gas): \(v_{avg} = \sqrt{\frac{8RT}{\pi M}}\) Where \(v_{avg}\) is the average molecular speed, \(R\) is the gas constant (8.314 J/mol·K), \(T\) is the temperature (300 K), and \(M\) is the molar mass of the gas. Since we're only comparing speeds relative to each other, we can ignore the constant terms and only compare the inverse of the square root of molar masses. 1. CO: \(\frac{1}{\sqrt{M_{\mathrm{CO}}}}\) 2. SF6: \(\frac{1}{\sqrt{M_{\mathrm{SF_{6}}}}}\) 3. H2S: \(\frac{1}{\sqrt{M_{\mathrm{H_{2} S}}}\) 4. Cl2: \(\frac{1}{\sqrt{M_{\mathrm{Cl_{2}}}}}\) 5. HBr: \(\frac{1}{\sqrt{M_{\mathrm{HBr}}}}\) Arranging them in increasing order (taking the inverse of the square root of molar masses): \(SF_{6} < HBr < Cl_{2} < H_{2}S < CO\)

Step 3: Calculate the rms Speeds

Now we'll calculate the rms speeds of CO and Cl₂ at 300 K, using the formula: \(v_{rms} = \sqrt{\frac{3RT}{M}}\) For CO: \(v_{\mathrm{rms} (\mathrm{CO})} = \sqrt{\frac{3 \cdot 8.314 \cdot 300}{0.02801}} = 430.50 \ \mathrm{m/s}\) For Cl₂: \(v_{\mathrm{rms} (\mathrm{Cl_{2})} = \sqrt{\frac{3 \cdot 8.314 \cdot 300}{0.07090}} = 301.59 \ \mathrm{m/s}\)

Step 4: Calculate the Most Probable Speeds

Lastly, we'll calculate the most probable speeds of CO and Cl₂ at 300 K, using the formula: \(v_{p} = \sqrt{\frac{2RT}{M}}\) For CO: \(v_{\mathrm{p} (\mathrm{CO})} = \sqrt{\frac{2 \cdot 8.314 \cdot 300}{0.02801}} = 384.91 \ \mathrm{m/s}\) For Cl₂: \(v_{\mathrm{p} (\mathrm{Cl_{2})} = \sqrt{\frac{2 \cdot 8.314 \cdot 300}{0.07090}} = 269.73 \ \mathrm{m/s}\) To summarize: (a) Order of increasing average molecular speed: \(SF_{6} < HBr < Cl_{2} < H_{2}S < CO\) (b) rms speeds: \(v_{\mathrm{rms} (\mathrm{CO})} = 430.50 \ \mathrm{m/s}\) \(v_{\mathrm{rms} (\mathrm{Cl_{2})} = 301.59 \ \mathrm{m/s}\) (c) Most probable speeds: \(v_{\mathrm{p} (\mathrm{CO})} = 384.91 \ \mathrm{m/s}\) \(v_{\mathrm{p} (\mathrm{Cl_{2})} = 269.73 \ \mathrm{m/s}\)

Key Concepts

Average Molecular Speed

Understanding the concept of average molecular speed is essential for grasping the behavior of gases. This speed represents the mean velocity of a collection of gas molecules at a given temperature. Simply, it's a way to describe how fast the molecules in a gas are moving on average.

While dealing with gases, temperature and molecular weight play pivotal roles in determining molecular speeds. In general, for gases at the same temperature but with different molar masses, the lighter gas molecules will have higher average speeds. This is in line with the kinetic molecular theory, which also implies that as the temperature increases, the average molecular speed increases as well.

The average molecular speed is a statistical figure that helps predict the rate of diffusion, effusion, and even the rate of reaction for gases. By mastering this concept, students can better understand the dynamic nature of gases and their interactions.

Root-Mean-Square (rms) Speed

The root-mean-square speed provides a quantitative measure of the speed of particles in a gas. This value is particularly useful as it relates to the kinetic energy of the gas particles. The rms speed is calculated using the formula:

\[\begin{equation}v_{rms} = \sqrt{\frac{3RT}{M}}\end{equation}\]
where \(v_{rms}\) is the root-mean-square speed, \(R\) represents the gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas in kilograms per mole.

Given that rms speed takes into account the distribution of speeds, it isn't the speed of any one molecule but instead gives an average speed based on the kinetic energy distribution. Understanding rms speed is critical for calculations involving kinetic energy and interpreting the physical implications of changes in temperature and molar mass on gas molecules.

Most Probable Speed

Moving on to the most probable speed, it can be conceptualized as the speed possessed by the largest number of molecules in a gas sample. Unlike the average or rms speeds, the most probable speed doesn't account for the mass of molecules but gives us a peak into the mode of the speed distribution curve, known as the Maxwell-Boltzmann distribution.

The formula to calculate this speed is as follows:

\[\begin{equation}v_{p} = \sqrt{\frac{2RT}{M}}\end{equation}\]
This equation showcases that higher temperatures and lower molar masses result in greater most probable speeds. This speed is paramount when discussing the escape velocity of molecules from a surface or when contemplating the likelihood of molecular collisions.

Molar Mass Calculation

Calculating molar mass is a fundamental skill in chemistry. The molar mass is essentially the weight of one mole of a substance, which is the sum of the atomic masses of that substance's constituent atoms. The unit for molar mass is grams per mole (g/mol), and it plays a considerable role in determining the properties of a gas, like its average, rms, and most probable speeds.

For each compound, the molar mass is calculated by adding together the standard atomic weights of the atoms present in the molecular formula. This is a critical step in many chemistry problems and requires the understanding of periodic table entries. Accurate molar mass calculation is vital for subsequent tasks, such as stoichiometric conversions and gas speed calculations.

Kinetic Molecular Theory

The kinetic molecular theory is a cornerstone of understanding gaseous behavior. It describes how gas molecules move and interact, stating that gas consists of small particles in constant random motion. This theory explains the properties of ideal gases, including the relationships between pressure, volume, temperature, and the number of particles.

• Gases consist of particles that are in constant, random motion.
• The particles are small and the distances between them are relatively large.
• Collisions between gas particles and their container walls are elastic, meaning there is no net loss of kinetic energy.
• The average kinetic energy of a collection of gas particles is proportional to the temperature of the gas in Kelvin.

Understanding this theory helps clarify why certain gases move faster and how energy is transferred between particles, influencing calculations of speed and behavior under various conditions.

Gas Constant (R)

The gas constant, known as \(R\), is fundamental in the study of gas laws. It appears in many fundamental equations, including the ideal gas law and the aforementioned speed equations. The value of \(R\) in SI units is 8.314 J/mol·K, but it can take on different values when using different units.

The gas constant bridges the gap between the microscopic and macroscopic worlds by relating the individual motion of gas particles to the pressure, volume, and temperature observable at the macro level. In calculations of molecular speeds or when applying the ideal gas law, \(R\) serves as a constant that allows us to use temperature, pressure, and volume to infer other properties of the gas. Its value is uniform for all ideal gases, making it a universal parameter in gas-related calculations.