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

CH₂F₂, Ar, N₂, CH₄

Step 1: Understand Root Mean Square Speed Formula

The root mean square speed (rms speed) of a gas molecule is given by the formula \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the gas constant (8.314 J/mol·K), \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas in kilograms per mole. At a constant temperature, \( v_{rms} \) depends inversely on the square root of the molar mass, \( M \).

Step 2: Convert Temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin. The temperature is \( 25^{\circ} \mathrm{C} \). Convert this to Kelvin with the formula: \[ T (K) = 25 + 273.15 = 298.15 \, \mathrm{K} \]

Step 3: Determine Molar Masses of Each Gas

Calculate the molar mass of each gas in kg/mol:- **Ar**: 39.95 g/mol or 0.03995 kg/mol.- **CH₄**: \(12.01 + 4(1.01) = 16.05\) g/mol or 0.01605 kg/mol.- **N₂**: \(2(14.01) = 28.02\) g/mol or 0.02802 kg/mol.- **CH₂F₂**: \(12.01 + 2(1.01) + 2(19.00) = 52.02\) g/mol or 0.05202 kg/mol.

Step 4: Compare Molar Masses

Arrange the gases by their molar masses from lightest to heaviest, as this determines the order of their increasing \( v_{rms} \). The lighter the molar mass, the higher the \( v_{rms} \).1. CH₄ (0.01605 kg/mol)2. N₂ (0.02802 kg/mol)3. Ar (0.03995 kg/mol)4. CH₂F₂ (0.05202 kg/mol)

Step 5: Arrange Gases by Increasing RMS Speed

The gas with the smallest molar mass has the highest \( v_{rms} \), and the one with the largest molar mass has the lowest \( v_{rms} \). Thus, the order of gases from lowest to highest \( v_{rms} \) is:1. CH₂F₂2. Ar3. N₂4. CH₄

Key Concepts

Root Mean Square Speed

The root mean square speed (rms speed) is a measure that helps to understand how fast molecules are moving in a gas. This concept comes from kinetic molecular theory, which describes gas molecules as constantly moving. The formula for rms speed is expressed as \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) represents the universal gas constant (8.314 J/mol·K), \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas in kilograms per mole.
The rms speed reveals that as the temperature increases, the speed of the gas molecules also increases. Conversely, heavier molecules, with greater molar mass \( M \), will move more slowly as part of a gas mixture. Thus, at a constant temperature, gases with lower molar masses will move faster than gases with higher molar masses. This direct relationship between temperature and speed, coupled with the inverse relationship between molar mass and speed, is crucial for understanding gas behavior.

Molar Mass

Molar mass is the mass of a given substance (chemical element or compound) divided by the amount of substance, usually expressed in grams per mole (g/mol). It's important in understanding how gases behave because it affects how quickly gas molecules move. For elements, this value is dictated by the periodic table, while for compounds, we sum the atomic masses of all atoms in the molecular formula.
For example, let's break down the molar mass for methane (\( CH_4 \)). The carbon atom has a molar mass of 12.01 g/mol, and the four hydrogen atoms each have a molar mass of about 1.01 g/mol. Thus, the total molar mass for methane is \( 12.01 + 4 \times 1.01 = 16.05 \) g/mol. When comparing gases like argon, methane, nitrogen, and difluoromethane, those with lower molar masses will have higher root mean square speeds at the same temperature, as their molecules can move more rapidly.

Gas Laws

Gas laws offer a framework for understanding how gas particles interact with each other and their surroundings. These laws include principles like Boyle's Law, Charles's Law, and Avogadro's Law.

• Boyle’s Law states that the pressure of a gas inversely relates to its volume when temperature is constant.
• Charles’s Law says that the volume of a gas is directly proportional to its temperature, assuming pressure remains unchanged.
• Avogadro’s Law shows the relationship between the size of gas particles and gas volume, emphasizing that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules.

These laws are unified in the ideal gas law equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature. While the ideal gas law is idealized, it offers insight into how gases behave under various conditions, helping predict movements and changes in gases discussed in concepts like root mean square speed and temperature changes.