Problem 45 - Chemistry and Chemical Reactivity

Question

Place the following gases in order of increasing rms speed at \(25^{\circ} \mathrm{C}: \mathrm{Ar}, \mathrm{CH}_{4}, \mathrm{N}_{2}, \mathrm{CH}_{2} \mathrm{F}_{2}\)

Step-by-Step Solution

Answer

Order of increasing rms speed: \(\mathrm{CH}_2\mathrm{F}_2\), \(\mathrm{Ar}\), \(\mathrm{N}_2\), \(\mathrm{CH}_4\).

Step 1: Understand the Concept of rms Speed

The root-mean-square (rms) speed of a gas is given by the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the molar mass of the gas in kilograms per mole. This formula tells us that rms speed inversely depends on the molar mass of the gas – the lighter the gas, the higher the speed.

Step 2: Convert Temperature to Kelvin

The temperature given is \(25^{\circ}\mathrm{C}\). To convert this to Kelvin, use the formula \(T(K) = T(^{\circ}C) + 273.15\). This gives us \(25 + 273.15 = 298.15\, \mathrm{K}\).

Step 3: Calculate the Molar Mass of Each Gas

The molar masses are as follows: \(\mathrm{Ar} = 39.95 \ \mathrm{g/mol}\), \(\mathrm{CH}_4 = 16.04 \ \mathrm{g/mol}\), \(\mathrm{N}_2 = 28.02 \ \mathrm{g/mol}\), and \(\mathrm{CH}_2\mathrm{F}_2 = 52.02 \ \mathrm{g/mol}\).

Step 4: Rank the Gases by Molar Mass

Arrange the gases by their molar mass in ascending order: \(\mathrm{CH}_4\) (16.04), \(\mathrm{N}_2\) (28.02), \(\mathrm{Ar}\) (39.95), \(\mathrm{CH}_2\mathrm{F}_2\) (52.02). Since rms speed is inversely proportional to the square root of molar mass, this order corresponds to increasing rms speed.

Step 5: List the Gases in Order of Increasing rms Speed

Using the logic that lower molar mass results in higher rms speed, we can now list the gases in order of increasing rms speed: \(\mathrm{CH}_2\mathrm{F}_2\), \(\mathrm{Ar}\), \(\mathrm{N}_2\), \(\mathrm{CH}_4\).

Key Concepts

Molar Mass Influence

Molar mass is a key factor in determining the root-mean-square (rms) speed of gases. The molar mass is the weight of the gas molecule in grams per mole. According to the formula for rms speed, \( v_{rms} = \sqrt{\frac{3kT}{m}} \), the rms speed is inversely proportional to the square root of the molar mass. This means that gases with lower molar masses move faster, while those with higher molar masses move more slowly.

For instance, in the given problem, methane \( \mathrm{CH}_4 \) has the lowest molar mass of 16.04 g/mol. This means it will travel fastest among the listed gases. Conversely, difluoromethane \( \mathrm{CH}_2\mathrm{F}_2 \) with a molar mass of 52.02 g/mol, being the heaviest, will have the slowest rms speed.

• Lighter gases have faster rms speeds.
• Heavier gases have slower rms speeds.

This concept helps in predicting and comparing the speeds of different gases under identical conditions.

Temperature Conversion to Kelvin

Temperature plays a crucial role in calculating the rms speed, and it is vital to express it in Kelvin for accurate calculations. The Kelvin scale starts at absolute zero, the point at which all molecular motion stops, making it the standard for scientific calculations.

To convert a temperature from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. For the problem's given temperature of \(25^\circ \mathrm{C}\), the conversion gives:
\[ T(K) = 25 + 273.15 = 298.15 \, \mathrm{K} \]This provides the absolute temperature, which ensures the calculations for rms speed accurately reflect the kinetic energy of the gas molecules. Using Kelvin is essential as it avoids negative temperatures, enabling precise scientific evaluations.

RMS Speed Calculation

The computation of the root-mean-square (rms) speed involves determining the average velocity of gas molecules in a sample. The equation for rms speed, \( v_{rms} = \sqrt{\frac{3kT}{m}} \), incorporates:

• \(k\) - Boltzmann constant (1.38 \times 10^{-23} \text{ J/K})
• \(T\) - Temperature in Kelvin
• \(m\) - Molar mass in kg/mol (be sure to convert g/mol to kg/mol)

The rms speed provides an insight into how fast gas molecules move and is crucial in many scientific and engineering applications. Each variable in the equation directly affects the result. The temperature (expressed in Kelvin) directly influences the speed, while the molar mass inversely affects it. This reveals an inverse relationship: as molar mass increases, speed decreases, and vice versa.

Gas Speed Ranking

When determining the order of gases by their rms speeds, the ranking depends primarily on their molar masses. The inverse proportionality means that the less massive a gas molecule, the faster it will move.

• Begin by listing gases based on their molar mass from lightest to heaviest.
• Then, rank the gases from slowest to fastest rms speed by reversing this list.

In the example, using the molar masses:

• \( \mathrm{CH}_4 \) - 16.04 g/mol
• \( \mathrm{N}_2 \) - 28.02 g/mol
• \( \mathrm{Ar} \) - 39.95 g/mol
• \( \mathrm{CH}_2\mathrm{F}_2 \) - 52.02 g/mol

The resulting rms speed ranking from slowest to fastest would be: \( \mathrm{CH}_2\mathrm{F}_2, \mathrm{Ar}, \mathrm{N}_2, \mathrm{CH}_4 \). This methodology clearly shows how a better understanding of molar mass can make speed predictions more accurate in chemical and physics applications.