Problem 22 - Chemistry in 30 days

Question

Which of the following is arranged in order of increasing molecular speed? (a) \(\mathrm{HBr}<\mathrm{O}_{2}<\mathrm{N}_{2}<\mathrm{H}_{2}\) (b) \(\mathrm{H}_{2}<\mathrm{N}_{2}<\mathrm{O}_{2}<\mathrm{HBr}\) (c) \(\mathrm{O}_{2}<\mathrm{HBr}<\mathrm{N}_{2}<\mathrm{H}_{2}\) (d) \(\mathrm{N}_{2}<\mathrm{O}_{2}<\mathrm{HBr}<\mathrm{H}_{2}\)

Step-by-Step Solution

Answer

Option (a) \( \mathrm{HBr}<\mathrm{O}_{2}<\mathrm{N}_{2}<\mathrm{H}_{2} \).

Step 1: Understand Molecular Speed

Molecular speed is inversely related to the square root of the molar mass, as defined by the equation \( v = \sqrt{\frac{3RT}{M}} \), where \( v \) is the average speed, \( R \) is the gas constant, \( T \) is temperature, and \( M \) is molar mass. Lighter molecules move faster than heavier ones.

Step 2: Determine Molar Masses

Calculate the molar masses of the molecules: - \( \text{H}_2 \): Molar mass = 2 g/mol- \( \text{N}_2 \): Molar mass = 28 g/mol- \( \text{O}_2 \): Molar mass = 32 g/mol- \( \text{HBr} \): Molar mass = 80.9 g/molThe order of molar mass from highest to lowest is: \( \text{HBr} > \text{O}_2 > \text{N}_2 > \text{H}_2 \).

Step 3: Compare Molar Mass and Speed

Based on the relationship, the order of increasing speed is opposite to the order of increasing molar mass. Therefore, the order of increasing speed is: \( \text{HBr} < \text{O}_2 < \text{N}_2 < \text{H}_2 \).

Step 4: Match to Options

Look at the options and find which matches our derived order. Option (a) \( \text{HBr}<\text{O}_2<\text{N}_2<\text{H}_2 \) matches perfectly with the calculated order.

Key Concepts

Molar Mass

Molar mass is a key factor in determining the speed at which molecules travel. It refers to the mass of one mole of a given substance, usually measured in grams per mole (g/mol). To find the molar mass, you need to add up the atomic masses of all the atoms in a molecule. For instance, the molar mass of hydrogen (\(\text{H}_2\)) is calculated by summing the atomic masses of its two hydrogen atoms, which equals 2 g/mol.
The molar mass not only provides insight into the molecular composition but also considerably affects molecular behavior. In the context of molecular speed, there is an inverse relationship between mass and speed. Lighter molecules, with lower molar masses, tend to move faster as they experience less resistance.
When comparing molecules, knowing their molar masses helps us predict their speed at a given temperature. So, with the molecules given:

• \(\text{H}_2\): 2 g/mol
• \(\text{N}_2\): 28 g/mol
• \(\text{O}_2\): 32 g/mol
• \(\text{HBr}\): 80.9 g/mol

we know that\(\text{H}_2\), the lightest, will have the highest speed, and\(\text{HBr}\), the heaviest, will have the lowest speed.

Gas Constant

The gas constant, denoted as\(R\), plays a pivotal role in thermodynamics and kinetics. In the equation for molecular speed,\(v = \sqrt{\frac{3RT}{M}}\),\(R\)helps relate various gas properties. It is a constant value used to link pressure, volume, and temperature of a gas in equations, commonly having a value of 8.314 J/(mol K).
In the context of the average speed of molecules,\(R\)ensures that calculations are consistent under standard conditions. It's part of the formula that allows us to calculate how fast molecules are moving based on their molar mass and the surrounding temperature.Understanding\(R\)helps us comprehend how various gas properties interconnect and influence each other. Recognizing this interconnectivity allows for predicting how changes in temperature or pressure might affect molecular speed. This constant reinforces the principle that higher temperature generally increases molecular speed, given a constant molar mass.

Average Speed

Average speed is an essential concept in the study of gases. It indicates the typical speed at which molecules in a gas move. The formula used to determine the average speed is:\[v = \sqrt{\frac{3RT}{M}}\]where:

• \(v\) = average speed
• \(R\) = gas constant
• \(T\) = temperature in Kelvin
• \(M\) = molar mass

Understanding this concept is crucial because it helps us describe how molecules behave in different conditions. If you increase the temperature,\(T\), molecules gain energy and speed up. Likewise, if you consider a molecule with a lower molar mass,\(M\), like hydrogen compared to bromine, its average speed is higher because it experiences less inertia.
Grasping the average speed equips you with the ability to predict reactions and movements of gases across various scientific disciplines. As a trick, always remember that lighter gases at higher temperatures travel faster than heavier gases, which neatly fits into understanding molecular dynamics.