Problem 24 - Chemistry in 30 days

Question

In which of the following pairs of the molecules (gaseous) have the same root mean square speed ? (a) He and \(\mathrm{Ar}\) (b) \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\) (c) \(\mathrm{O}_{2}\) and \(\mathrm{O}_{3}\) (d) \(\mathrm{N}_{2}\) and \(\mathrm{CO}\)

Step-by-Step Solution

Answer

(d) N\( _2 \) and CO have the same root mean square speed.

Step 1: Understanding the Root Mean Square Speed Formula

The root mean square speed of a gas depends on the temperature and the molar mass of the gas. The formula to calculate the root mean square speed, \( u_{rms} \), is \( u_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass in kilograms per mole.

Step 2: Compare Molar Masses of Gaseous Molecules

We need to find the molecular weight in \( g/mol \) or \( kg/mol \) for each pair of molecules. Then, we can compare them because, at the same temperature, gases with the same molar mass have the same root mean square speed.

Step 3: Calculate Molar Mass of Given Molecules

Calculate the molar masses:- He: \( 4 \, \text{g/mol} \)- Ar: \( 40 \, \text{g/mol} \)- NO\( _2 \): \( 46 \, \text{g/mol} \)- N\( _2 \)O\( _4 \): \( 92 \, \text{g/mol} \)- O\( _2 \): \( 32 \, \text{g/mol} \)- O\( _3 \): \( 48 \, \text{g/mol} \)- N\( _2 \): \( 28 \, \text{g/mol} \)- CO: \( 28 \, \text{g/mol} \).

Step 4: Identify Pairs with the Same Molar Mass

From the calculations, only N\( _2 \) and CO both have a molar mass of \( 28 \, \text{g/mol} \); since their molar mass is the same, their root mean square speed at the same temperature will also be the same.

Key Concepts

Molar Mass

Molar mass refers to the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is a fundamental concept in chemistry that links the mass of a substance to the amount in moles.
In the context of root mean square speed, molar mass plays a key role because it affects how fast or slow gas molecules move. Lighter molecules, with lower molar mass, generally move faster than heavier ones. For instance, helium (He) has a molar mass of 4 g/mol, making it much lighter and thus faster than a gas like argon (Ar), which has a molar mass of 40 g/mol.
Understanding molar mass is crucial for calculating and comparing the rates at which different gases move, especially when using equations such as the root mean square speed formula.

Gas Constant

The gas constant, often symbolized as \( R \), is a key component in the root mean square speed formula. The value of \( R \), given as 8.314 J/mol·K, is a universal constant appearing in various equations related to gases, including the Ideal Gas Law.
In the formula for root mean square speed \( u_{rms} = \sqrt{\frac{3RT}{M}} \), the gas constant sets a proportional relationship between temperature, molar mass, and the speed of gas molecules. It helps balance these variables, showing us how changes in one can affect the others. For instance, a higher temperature will increase \( u_{rms} \) since temperature and speed are directly related through \( R \).
Grasping the role of the gas constant is essential for understanding how interconnected the physical properties of gases are.

Temperature in Kelvin

The temperature in Kelvin scales directly with the kinetic energy of gas molecules. When working with physical formulas, Kelvin is preferred over Celsius because it starts at absolute zero, the point where all molecular motion ceases. This makes it a true measure of thermal energy.
In exercises involving the root mean square speed, temperature must be in Kelvin to maintain the integrity of the gas laws. Colder temperatures result in slower molecular speeds, while warmer temperatures boost energy and speed. For example, gases at 300 K will move faster than those at 273 K.
By using temperature in Kelvin, we ensure calculations reflect the actual physicochemical behavior of gas molecules.

Comparative Molar Mass Analysis

Comparative molar mass analysis involves comparing the molar masses of different gases to predict their behavior under similar conditions. When gases have the same molar mass, they exhibit the same velocity at the same temperature according to the root mean square speed formula.

• If two gases have the same molar mass, they will have the same root mean square speed at identical temperatures. An example is nitrogen (\( N_2 \)) and carbon monoxide (CO), both with a molar mass of 28 g/mol.
• This comparative analysis helps identify gases with similar physical characteristics, aiding in predictions about how they will react or behave.
• Practicing this analysis enhances understanding of molecular kinetics and the effects of mass on molecular speed.

Recognizing these similarities can be vital in fields like industrial chemistry where gas reaction rates are crucial.