Question

Two scaled containcrs of the samc capacity and at the same tempcraturc arc filled with \(44 \mathrm{~g}\) of \(\mathrm{II}_{2}\) gas in onc and \(44 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) in the other. If the \(P\) of \(\mathrm{CO}_{2}\) is \(1 \mathrm{~atm}\) in other the \(P\) of \(\mathrm{II}_{2}\) in its container will be (1) 1 atm (2) zero (3) \(22 \mathrm{~atm}\) (4) 4 atm

Short answer

22 atm

Step-by-step solution

- Determine Molar Mass

Calculate the molar mass of both gases. The molar mass of \(\text{H}_2\) is 2 g/mol and the molar mass of \(\text{CO}_2\) is 44 g/mol.

- Calculate Number of Moles

Find the number of moles of each gas. Use the formula \(\text{number of moles} = \frac{\text{mass}}{\text{molar mass}}\)For \(\text{H}_2\): \(n(\text{H}_2 ) = \frac{44 \text{ g}}{2 \text{ g/mol}} = 22 \text{ moles}\)For \(\text{CO}_2\): \(n(\text{CO}_2) = \frac{44 \text{ g}}{44 \text{ g/mol}} = 1 \text{ mole}\)

- Use Ideal Gas Law

Apply the ideal gas law, \(PV = nRT\), assuming same volume, temperature, and gas constant (R) for both gases. Since P = nRT/V, pressure is proportional to the number of moles of the gas when V and T are constant. Given \(P_{\text{CO}_2} = 1 \text{ atm}\):Set \(P_{\text{II}_2} = 22 \text{ atm}\) because it has 22 times more moles.

Key concepts

Molar Mass Calculation

Understanding molar mass is crucial to solving problems related to gases. Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). For example, Hydrogen gas \(\text{H}_2\), has a molar mass of 2 g/mol because each hydrogen atom has a molar mass of 1 g/mol, and there are two atoms. For Carbon Dioxide \(\text{CO}_2\), the molar mass is 44 g/mol (12 g/mol for carbon and 32 g/mol for the two oxygen atoms combined).

To determine the molar mass of any gas:

• Find the molar mass of each constituent element
• Multiply the atomic mass by the number of each type of atom in the molecule
• Add these values together to get the total molar mass of the compound

This step lays the foundation for future calculations in most gas law problems.

Number of Moles

The number of moles, often represented as \(n\), is a measure of the amount of substance. To find it, you divide the given mass of the substance by its molar mass: \[ n = \frac{mass}{molar\text{ }mass} \]

For example, if you have 44 grams of Hydrogen gas (H2):

• The molar mass of H2 is 2 g/mol
• Thus, the number of moles \(\text{H}_2\) is \( \frac{44 \text{ g}}{2 \text{ g/mol}} = 22 \text{ moles} \)

Similarly, for 44 grams of Carbon Dioxide (CO2):

• The molar mass of CO2 is 44 g/mol
• Thus, the number of moles \( \text{CO}_2 \) is \(\frac{44 \text{ g}}{44 \text{ g/mol}} = 1 \text{ mole} \)

This concept is essential as it directly influences other properties like gas pressure.

Proportionality in Gas Laws

The Ideal Gas Law, \( PV = nRT \), is essential for connecting the physical properties of gases. Here, \(P \) is pressure, \(V \) is volume, \(n \) is the number of moles, \(R \) is the gas constant, and \(T \) is temperature.

In our example, the containers have the same volume and temperature. This means that the pressure of the gas can be directly linked to its number of moles, according to the formula:

• \( P = \frac{nRT}{V} \)
• Because \(T \), \(V \), and \(R \) are constant, the pressure is directly proportional to the number of moles: \( P \) is proportional to \( n \)

For instance, if \( \text{H}_2 \) has 22 times the number of moles compared to \( \text{CO}_2 \) in the same conditions, its pressure will be 22 times higher. Thus, given that the pressure of \( \text{CO}_2 \) is 1 atm, the pressure of \( \text{H}_2 \) will be 22 atm.

This principle is crucial for understanding and predicting the behavior of gases.