Question

Calculate the root mean square velocities of \(\mathrm{CH}_{4}(\mathrm{~g})\) and \(\mathrm{N}_{2}(\mathrm{~g})\) molecules at \(273 \mathrm{~K}\) and \(546 \mathrm{~K}\).

Short answer

The root mean square velocities of CH4(g) and N2(g) molecules at the given temperatures are: CH4(g) at 273 K: \(660.26 m/s\), CH4(g) at 546 K: \(934.98 m/s\), N2(g) at 273 K: \(490.93 m/s\), and N2(g) at 546 K: \(694.40 m/s\).

Step-by-step solution

Root Mean Square Velocity Formula

The formula for calculating the root mean square velocity (v_rms) of a gas molecule is given by: \[v_{rms} = \sqrt{\frac{3RT}{M}}\] where R is the ideal gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and M is the molar mass of the gas in kg/mol.

Molar Mass of CH4 and N2

We need to find the molar mass of CH4 and N2 in kg/mol to use in the formula. Molar mass of CH4: 12.01 g/mol (C) + 4 × 1.01 g/mol (H) = 16.04 g/mol = 0.01604 kg/mol Molar mass of N2: 2 × 14.01 g/mol (N) = 28.02 g/mol = 0.02802 kg/mol

Calculate v_rms for CH4 at 273 K

Plug in the values of R, T, and M for CH4 in the formula: \[v_{rms} = \sqrt{\frac{3 × 8.314 J/(mol·K) × 273 K}{0.01604 kg/mol}} = 660.26 m/s\]

Calculate v_rms for CH4 at 546 K

Plug in the values of R, T, and M for CH4 in the formula: \[v_{rms} = \sqrt{\frac{3 × 8.314 J/(mol·K) × 546 K}{0.01604 kg/mol}} = 934.98 m/s\]

Calculate v_rms for N2 at 273 K

Plug in the values of R, T, and M for N2 in the formula: \[v_{rms} = \sqrt{\frac{3 × 8.314 J/(mol·K) × 273 K}{0.02802 kg/mol}} = 490.93 m/s\]

Calculate v_rms for N2 at 546 K

Plug in the values of R, T, and M for N2 in the formula: \[v_{rms} = \sqrt{\frac{3 × 8.314 J/(mol·K) × 546 K}{0.02802 kg/mol}} = 694.40 m/s\] So, the root mean square velocities are as follows: CH4(g) at 273 K: 660.26 m/s CH4(g) at 546 K: 934.98 m/s N2(g) at 273 K: 490.93 m/s N2(g) at 546 K: 694.40 m/s

Key concepts

Ideal Gas Constant

The ideal gas constant, often represented by the symbol R, is a fundamental constant that appears in the ideal gas law: \( PV = nRT \), where P is pressure, V is volume, n is the amount of substance in moles, and T is temperature in Kelvin. The value of R in SI units is \(8.314 J/(mol\cdot K)\). This constant provides a proportionality factor between the energy scale and the temperature scale on a per mole basis.

The importance of the ideal gas constant becomes evident when we discuss gas behavior and properties. In the context of calculating root mean square velocities of gas molecules, R is instrumental in bridging the connection between the macroscopic conditions of temperature and the microscopic kinetic energies of gas particles. Thus, understanding R is crucial to comprehend wider gas dynamics beyond just this equation.

Molar Mass

Molar mass is another key term in calculations involving gases. It is defined as the mass of one mole of a substance and is expressed in grams per mole (g/mol) or kilograms per mole (kg/mol) depending on the context. For molecules like methane (CH₄) and nitrogen (N₂), molar mass takes into account the mass of all the atoms present in a single molecule of the gas.

Molar mass plays a crucial role in converting the macroscopic measurements (like those obtained from a balance) to microscopic moles of particles, and vice versa. It also influences the behavior of a gas, as seen in the calculation of root mean square velocity, where the molar mass indirectly affects how fast gas particles move on average at a given temperature.

Formula Derivation

The derivation of the root mean square velocity formula invokes principles from kinetic molecular theory, which describes the behavior of gas molecules. This theory states that the kinetic energy of gas particles is proportional to the temperature of the gas. By equating the kinetic energy expression \(\frac{1}{2}mv^{2}\) with the thermodynamic definition that includes R and T, we derive the formula for root mean square velocity:\[v_{rms} = \sqrt{\frac{3RT}{M}}\]

In this formula, the square root is taken because we deal with squares of velocities when calculating average values of kinetic energy. The factor of 3 comes from the three degrees of freedom of motion in space (x, y, z directions) for a monatomic ideal gas, as each contributes equally to the kinetic energy.

Gas Molecule Velocities

Gas molecule velocities are not uniform within a sample of gas; rather, they follow a distribution. The root mean square velocity calculated using the formula derived from kinetic theory provides an average measure of this velocity distribution.

The velocities of gas molecules impact properties like diffusion rate, effusion, and the rate of gas mixing. The temperature dependence of these velocities, as embodied in the formula, shows that higher temperatures result in greater average gas molecule speeds. This relationship is fundamental to understanding and predicting the behavior of gases under various conditions, including reactions involving gases.