Question

You have two pressure-proof steel cylinders of equal volume, one containing \(1.0 \mathrm{kg}\) of \(\mathrm{CO}\) and the other containing \(1.0 \mathrm{kg}\) of acetylene, \(\mathrm{C}_{2} \mathrm{H}_{2}\) (a) In which cylinder is the pressure greater at \(25^{\circ} \mathrm{C} ?\) (b) Which cylinder contains the greater number of molecules?

Short answer

(a) Acetylene cylinder; (b) Acetylene cylinder.

Step-by-step solution

Understanding Molar Mass

Firstly, we need to determine the molar mass of each gas. For carbon monoxide (\( \text{CO} \)), the molar mass is calculated as follows: \( \text{C: } 12.01 \, \text{g/mol} + \text{O: } 16.00 \, \text{g/mol} = 28.01 \, \text{g/mol} \). For acetylene (\( \text{C}_2\text{H}_2 \)), the molar mass is: \( \text{C: } 12.01 \, \text{g/mol} \times 2 + \text{H: } 1.01 \, \text{g/mol} \times 2 = 26.04 \, \text{g/mol} \).

Calculating Number of Moles

To find out the number of moles in each cylinder, use the formula: \( n = \frac{\text{mass}}{\text{molar mass}} \). For \( \text{CO} \): \( n_{\text{CO}} = \frac{1000 \, \text{g}}{28.01 \, \text{g/mol}} \approx 35.7 \, \text{mol} \). For \( \text{C}_2\text{H}_2 \): \( n_{\text{C}_2\text{H}_2} = \frac{1000 \, \text{g}}{26.04 \, \text{g/mol}} \approx 38.4 \, \text{mol} \).

Applying Ideal Gas Law

According to the ideal gas law \( PV = nRT \), pressure (\( P \)) is directly proportional to the number of moles (\( n \)) assuming volume (\( V \)) and temperature (\( T \)) are constant. Since \( n_{\text{C}_2\text{H}_2} > n_{\text{CO}} \), the pressure is greater in the acetylene (\( \text{C}_2\text{H}_2 \)) cylinder.

Determining Number of Molecules

The number of molecules in a gas is given by \( N = nN_A \), where \( N_A \) is Avogadro's number, \( 6.022 \times 10^{23} \, \text{mol}^{-1} \). Since the acetylene cylinder has more moles (\( n_{\text{C}_2\text{H}_2} > n_{\text{CO}} \)), it also contains more molecules.

Key concepts

Molar Mass

Molar Mass is the mass of one mole of a given substance and is commonly expressed in grams per mole (g/mol). To calculate the molar mass, we need to sum up the atomic masses of all the atoms present in the molecular formula of the substance.

For example, in carbon monoxide (CO), the molar mass is determined by adding the atomic mass of carbon (12.01 g/mol) and oxygen (16.00 g/mol), resulting in a molar mass of 28.01 g/mol. Similarly, for acetylene (C\(_2\)H\(_2\)), we multiply the atomic mass of carbon by 2 (since there are two carbon atoms) and add to that the atomic mass of hydrogen multiplied by 2, giving a molar mass of 26.04 g/mol.

• CO has a molar mass of 28.01 g/mol
• C\(_2\)H\(_2\) has a molar mass of 26.04 g/mol

This concept is crucial in solving problems involving the Ideal Gas Law or calculating the number of moles, providing the foundational understanding needed in gas-related equations.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry and is invaluable for understanding the composition of substances at the molecular level. This number, 6.022 × 10\(^{23}\) mol\(^{-1}\), represents the number of atoms, ions, or molecules contained in one mole of a substance.

In chemical calculations, such as determining the number of molecules in a given gas, Avogadro's number comes into play. When you know the number of moles of a substance, multiplying by Avogadro's number gives you the total number of molecules (or atoms or ions, depending on what's relevant).

• Avogadro's number = 6.022 × 10\(^{23}\) mol\(^{-1}\)
• Use Avogadro's number to convert between moles and molecules

This concept helps bridge the macroscopic and microscopic worlds of chemistry, providing a method to count particles in amounts that are usable in the lab.

Number of Moles

The Number of Moles indicates how many moles of a substance are present in a given mass. To calculate it, you divide the mass of the substance by its molar mass. This calculation is central to using the Ideal Gas Law, where the number of moles is a key variable that directly affects the calculations for pressure, volume, and temperature.

For instance, in the given cylinders problem, the number of moles for both CO and C\(_2\)H\(_2\) was calculated using the formula: \[ n = \frac{\text{mass}}{\text{molar mass}} \]

Applying this formula:

• The CO cylinder contains approximately 35.7 moles.
• The C\(_2\)H\(_2\) cylinder contains approximately 38.4 moles.

The calculation of moles helps determine not only how much of a substance is present in terms of its elementary entities but also directly informs predictions about the behavior of the substance when conditions like pressure or temperature change.