Question

A \(1.25-\mathrm{g}\) sample of \(\mathrm{CO}_{2}\) is contained in a \(750 .\) mI. flask at \(22.5^{\circ} \mathrm{C} .\) What is the pressure of the gas?

Short answer

The pressure of the gas is approximately 0.922 atm.

Step-by-step solution

Convert Temperature to Kelvin

To find the pressure of the gas using the ideal gas law, we need the temperature in Kelvin. The formula to convert Celsius to Kelvin is \( K = ^{\circ}C + 273.15 \). Here, the temperature is given as \(22.5^{\circ}C\). Therefore, the temperature in Kelvin is:\[T = 22.5 + 273.15 = 295.65 \, K\]

Convert Mass to Moles

We need the number of moles of \( \mathrm{CO}_2 \) to use the ideal gas law. The molar mass of \( \mathrm{CO}_2 \) is calculated as \(12.01 \, g/mol + 2 \times 16.00 \, g/mol = 44.01 \, g/mol\). Given the mass of \( \mathrm{CO}_2 \) is \(1.25 \, g\), the number of moles \( n \) can be calculated using the formula:\[n = \frac{1.25 \, g}{44.01 \, g/mol} \approx 0.0284 \, mol\]

Convert Volume to Liters

The volume of the flask is given in milliliters, but we need it in liters for the ideal gas law. Since \(1 \, L = 1000 \, mL\), convert \(750 \, mL\) to liters:\[750 \, mL = 0.750 \, L\]

Use Ideal Gas Law to Find Pressure

We will use the ideal gas law \( PV = nRT \) to find the pressure \( P \), where \( R \) is the ideal gas constant, \(0.0821 \, L\cdot atm/(mol \cdot K)\). Rearrange the equation to solve for \( P \):\[P = \frac{nRT}{V}\]Substitute the known values:\[P = \frac{0.0284 \, mol \times 0.0821 \, L\cdot atm/(mol \cdot K) \times 295.65 \, K}{0.750 \, L}\]\[P \approx 0.922 \, atm\]

Conclusion

The pressure of the gas inside the flask is approximately \(0.922 \, atm\).

Key concepts

Molar Mass of CO2

Understanding the concept of molar mass is crucial when working with gases such as carbon dioxide (\(\mathrm{CO}_2\)). The molar mass of \(\mathrm{CO}_2\) is derived from the sum of the atomic masses of its constituent elements. CO2 consists of one carbon atom and two oxygen atoms.

• The atomic mass of carbon is approximately \(12.01 \, g/mol\).
• The atomic mass of oxygen is approximately \(16.00 \, g/mol\). With two oxygen atoms, this amounts to \(2 \times 16.00 \, g/mol\).

Thus, the molar mass of \(\mathrm{CO}_2\) is \(12.01 + 2 \times 16.00 = 44.01 \, g/mol\).
This value is essential for converting grams of CO2 to moles, which is a key step in many calculations involving gases. Knowing how to find this molar mass allows you to compute the number of moles from a given sample of CO2.

Pressure Calculation

Pressure calculation for gases often utilizes the ideal gas law, which states \(PV = nRT\). This equation relates the pressure (\(P\)), volume (\(V\)), and temperature (\(T\)) of a gas in relation to the moles of gas (\(n\)) present. The constant \(R\) is the universal gas constant and usually has a value of \(0.0821 \, L \cdot atm/(mol \cdot K)\).

Pressure is a measure of the force the gas exerts on the walls of its container. By rearranging the ideal gas law formula to \(P = \frac{nRT}{V}\), you can solve for pressure.
This step requires values for moles, volume, and temperature in correct units.

• The temperature must be in Kelvin.
• The volume should be in liters.
• The number of moles is calculated from the mass and molar mass of the gas.

Substituting these known values into the ideal gas law formula gives you the pressure in atmospheres (\(atm\)). Correctly using the ideal gas law to calculate pressure is foundational for understanding gas behavior under various conditions.

Temperature Conversion

Temperature conversion is a fundamental step in gas law calculations. Each temperature in these equations must be in Kelvin because Kelvin is the absolute temperature scale that starts from absolute zero. To convert degrees Celsius to Kelvin, use the formula: \[K = ^{\circ}C + 273.15\]

Let's illustrate with an example. If the given temperature is \(22.5^{\circ} C\), convert it as follows:

• \(^\circ C = 22.5\)
• Add \(273.15\) to the Celsius temperature.
• \(K = 22.5 + 273.15 = 295.65 \, K\)

This conversion allows the temperature to be used in the ideal gas law. Always ensure temperatures are in Kelvin when using this law, as it directly relates to the kinetic energy of the molecules involved.

Volume Conversion

In chemistry, it's crucial to use consistent units, especially when applying the ideal gas law \(PV = nRT\). For gases, volume must be expressed in liters. Hence, converting milliliters to liters is necessary. The conversion factor is straightforward: \[1 \, L = 1000 \, mL\]

To convert a given volume of gas from milliliters to liters, simply divide by 1000. For instance, if you have a flask with \(750 \, mL\) of gas, the conversion should be calculated as follows:

• \(Volume \, in \, mL = 750\, mL\)
• Convert to liters: \(0.750 \, L = \frac{750}{1000}\)

Maintaining this unit consistency is vital for correctly applying the ideal gas law, as discrepancies in units can lead to errors in pressure, temperature, or various measurements. Properly converted volumes ensure that the subsequent pressure calculations are accurate.