Question

Bottled gas for recreational vehicles contains propane, \(\mathrm{C}_{3} \mathrm{H}_{8},\) and butane, \(\mathrm{C}_{4} \mathrm{H}_{10} .\) Which molecules have the faster velocity?

Short answer

Propane molecules have the faster velocity.

Step-by-step solution

Identify the Gases

First, identify the two gases involved in this problem: propane ( C_3H_8 ) and butane ( C_4H_10 ). We need to consider their molecular weights to determine which gas molecules have the faster velocity.

Calculate Molecular Weights

Calculate the molar mass of each gas: for propane ( C_3H_8 ), it is 3 imes 12.01 ext{ g/mol} + 8 imes 1.008 ext{ g/mol} = 44.09 ext{ g/mol} , and for butane ( C_4H_10 ), it is 4 imes 12.01 ext{ g/mol} + 10 imes 1.008 ext{ g/mol} = 58.12 ext{ g/mol} .

Use Kinetic Molecular Theory

According to the kinetic molecular theory, at the same temperature, the average kinetic energy of different gases is the same. Since kinetic energy E_k is proportional to 1/2 imes ext{mass} imes ext{velocity}^2 , lighter molecules will have higher velocity.

Compare Velocities

Using the inverse relationship between mass and velocity (v propto 1/ ext{mol.mass} ) for gases with the same kinetic energy, compare the velocities of the two gases. Since propane has a lower molar mass than butane, propane molecules have a higher velocity than butane molecules.

Key concepts

Propane and Butane Molecular Weights

To compare the velocities of propane and butane molecules, we first need to understand the concept of molecular weight, also known as molar mass. Molecular weight is essentially the sum of the atomic weights of all atoms in a molecule and is expressed in grams per mole (g/mol). For propane \( (\mathrm{C}_3\mathrm{H}_8) \), the calculation of molecular weight involves multiplying the number of carbon atoms by the atomic weight of carbon (\(12.01 \text{ g/mol} \)) and the number of hydrogen atoms by the atomic weight of hydrogen (\(1.008 \text{ g/mol} \)): - Carbons: \(3 \times 12.01 \text{ g/mol} = 36.03 \text{ g/mol} \) - Hydrogens: \(8 \times 1.008 \text{ g/mol} = 8.064 \text{ g/mol} \) Adding these gives us the total molecular weight of propane: \(44.09 \text{ g/mol} \). For butane \( (\mathrm{C}_4\mathrm{H}_{10}) \), the process is similar: - Carbons: \(4 \times 12.01 \text{ g/mol} = 48.04 \text{ g/mol} \) - Hydrogens: \(10 \times 1.008 \text{ g/mol} = 10.08 \text{ g/mol} \) So, the molecular weight of butane is \(58.12 \text{ g/mol} \). These calculations reveal that butane is heavier than propane.

Gas Velocity Comparison

The velocity of gas molecules can be compared using principles from the kinetic molecular theory. According to this theory, the average kinetic energy of gases at a given temperature is the same, regardless of the gas type. Kinetic energy \( (E_k) \) can be expressed as \(1/2 \times \text{mass} \times \text{velocity}^2\). This implies that lighter molecules, having less mass, must move faster to have the same kinetic energy as heavier molecules, which move slower. For propane and butane, where: - Propane has a molecular weight of \(44.09 \text{ g/mol} \) - Butane has a molecular weight of \(58.12 \text{ g/mol} \) Applying the relationship \( v \propto 1/\sqrt{\text{mol.mass}} \), where \(v\) is velocity, we can deduce that propane molecules will travel faster than butane molecules. This is because the lighter propane, with its lower molecular weight, needs to speed up more to balance the kinetic energy with that of the heavier butane.

Mole Concept Calculations

The mole concept is key in dealing with chemical calculations, as it relates amounts of substances to their molecular or atomic scale. One mole of any substance contains Avogadro's number of entities, which is \(6.022 \times 10^{23}\) atoms, molecules, or particles.When dealing with gases, the mole concept allows us to quantify the amount of substance in terms of Avogadro's constant. This, coupled with molar mass, helps in comparing gases like propane and butane. The molar mass directly tells us the mass of one mole of a substance. To see this in action: - One mole of propane with a molar mass of \(44.09 \text{ g/mol} \) means that \(44.09 \text{ g} \) of propane is equivalent to \(6.022 \times 10^{23}\) molecules. - Similarly, for butane, \(58.12 \text{ g/mol} \) means \(58.12 \text{ g} \) corresponds to the same number of molecules. Thus, using mole concept calculations, we can not only calculate amounts in chemical reactions but also predict behaviors and properties of gases based on their molecular weights.