Question

An aerosol spray can with a volume of \(350 \mathrm{~mL}\) contains \(3.2 \mathrm{~g}\) of propane gas \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) as propellant. What is the pressure in atmospheres of gas in the can at \(20{ }^{\circ} \mathrm{C}\) ?

Short answer

The pressure in the can is 4.92 atm.

Step-by-step solution

Convert Volume to Liters

First, convert the volume of the can from milliliters to liters. Since there are 1000 mL in a liter, use the conversion factor:\[ 350 \text{ mL} = 0.350 \text{ L} \]

Use Ideal Gas Law Formula

To find the pressure, apply the ideal gas law, which is expressed as:\[ PV = nRT \]where \( P \) is pressure, \( V \) is volume, \( n \) is number of moles, \( R \) is the universal gas constant \((0.0821 \frac{L \, atm}{mol \, K})\), and \( T \) is temperature in Kelvin. You'll need to find \( n \) and \( T \) next.

Convert Temperature to Kelvin

The given temperature is in Celsius, so convert it to Kelvin by adding 273:\[ T = 20 + 273 = 293 \text{ K} \]

Find Number of Moles

Calculate the number of moles of propane using its molar mass. The molar mass of propane (\(C_3H_8\)) is calculated as:\[ 3(12.01) + 8(1.01) = 44.11 \frac{g}{mol} \]Calculate the moles of propane:\[ n = \frac{3.2 \, g}{44.11 \, \frac{g}{mol}} = 0.0725 \, mol \]

Calculate Pressure Using Ideal Gas Law

Substitute the known values into the ideal gas law equation:\[ P = \frac{nRT}{V} = \frac{0.0725 \, mol \times 0.0821 \frac{L \, atm}{mol \, K} \times 293 \, K}{0.350 \, L} \]Calculate the result:\[ P = 4.92 \, atm \]

Key concepts

Pressure Calculation Using the Ideal Gas Law

Pressure calculation is an essential aspect of understanding gases, and the ideal gas law is a fundamental equation used in this context. The formula for the ideal gas law is given by \[ PV = nRT \]where:

• P is the pressure of the gas.
• V is the volume of the gas.
• n is the number of moles of the gas.
• R is the universal gas constant.
• T is the temperature in Kelvin.

To calculate pressure, rearrange the formula to solve for \[ P = \frac{nRT}{V} \]This equation tells us that pressure is directly proportional to the number of moles of gas and the temperature, but inversely proportional to the volume of the gas. In the exercise, knowing the amount of gas (in moles), the temperature (converted to Kelvin), and the volume (converted to liters) allowed for an accurate calculation of pressure in atmospheres. This approach is practical in various scientific and real-world applications, from chemistry labs to engineering problems.

Finding the Molar Mass of Propane

The molar mass of a compound provides vital information about the substance’s molecular weight, measured in grams per mole. For propane \((\text{C}_3\text{H}_8)\), its molar mass is calculated by considering the atomic masses of all the atoms in a molecule. Propane consists of three carbon atoms and eight hydrogen atoms. Thus, its molar mass can be expressed as:\[ 3 \times 12.01 + 8 \times 1.01 = 44.11 \, \text{g/mol} \]Understanding the molar mass is crucial when converting grams of a substance to moles, which is necessary for solving equations using the ideal gas law. This information bridges the gap between the macroscopic scale (grams) and the molecular scale (moles), helping to simplify calculations and provide meaningful interpretations in chemistry. By dividing the mass of propane given in the problem by its molar mass, one determines the number of moles present, which is a crucial step for pressure calculations.

Temperature Conversion to Kelvin

Temperature plays a critical role in gas law calculations, and the Kelvin scale is the absolute temperature scale used in these calculations. The conversion from Celsius to Kelvin is straightforward; simply add 273 to the Celsius temperature:\[ T_{\text{Kelvin}} = T_{\text{Celsius}} + 273 \]This conversion is necessary because the Kelvin scale starts at absolute zero, the theoretical temperature where molecular motion ceases. The Kelvin scale ensures that temperatures in gas law problems are always positive, which is necessary for mathematical reasons. In the exercise, the given temperature of 20°C was converted to 293 K. The Kelvin temperature is then used in the ideal gas law equation to calculate the pressure of the gas. Understanding this conversion is crucial since it applies to many scientific calculations, ensuring consistent and accurate results across different scientific disciplines.