Question

An aerosol spray can of deodorant 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 \(\mathrm{Pa}\) ) in the can at \(20^{\circ} \mathrm{C}\) ?

Short answer

The pressure in the can is approximately \(506,000 \, \mathrm{Pa}\).

Step-by-step solution

Identify the Known Values

Extract the given information from the problem statement. The volume of the can is given as \(V=350 \mathrm{~mL}\). Convert this to cubic meters: \(V=350 \times 10^{-6} \mathrm{~m}^3\). The mass of propane is \(3.2 \, \mathrm{g}\). The temperature is \(20^{\circ} \mathrm{C}\), which we will convert to Kelvin: \(T = 20 + 273.15 = 293.15 \, \mathrm{K}\).

Calculate Moles of Propane

Use the molar mass of propane \(\mathrm{C}_3\mathrm{H}_8\), which is approximately \(44.1 \, \mathrm{g/mol}\), to determine the number of moles \(n\). \(n = \frac{3.2 \, \mathrm{g}}{44.1 \, \mathrm{g/mol}} \approx 0.0725 \, \mathrm{mol}\).

Use Ideal Gas Law

The ideal gas law is \(PV = nRT\). We know \(P\) is pressure, \(V = 350 \times 10^{-6} \, \mathrm{m}^3\), \(n \approx 0.0725 \, \mathrm{mol}\), and \(T = 293.15 \, \mathrm{K}\). The gas constant \(R\) is \(8.314 \, \mathrm{J/(mol \, K)}\).

Solve for Pressure

Rearrange the ideal gas law formula to solve for \(P\): \(P = \frac{nRT}{V}\). Substitute the known values: \(P = \frac{0.0725 \times 8.314 \times 293.15}{350 \times 10^{-6}}\).

Calculate and Convert to Pascals

Calculate the pressure: \(P \approx \frac{0.0725 \times 8.314 \times 293.15}{0.00035} = 505,994 \, \mathrm{Pa}\). Therefore, the pressure in the can is approximately \(506,000 \, \mathrm{Pa}\).

Key concepts

Pressure Calculation

Understanding how to calculate pressure involves using the Ideal Gas Law equation, which is \(PV = nRT\). Here, \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

To find pressure \(P\), rearrange the equation to \(P = \frac{nRT}{V}\). Simply put:

• Identify your known values: moles \(n\), temperature \(T\), and volume \(V\).
• Substitute into the equation and solve for \(P\).

In this case, we used the given values to find that the pressure was approximately \(506,000 \, \mathrm{Pa}\). This result helps us understand the force exerted by the gas in the container.

Gas Volume Conversion

Converting gas volume is crucial because the Ideal Gas Law requires the volume in cubic meters (\(\mathrm{m}^3\)), not milliliters (\(\mathrm{mL}\)).

To convert from milliliters to cubic meters, use the conversion:

• 1 \(\mathrm{mL}\) is equal to \(1 \times 10^{-6} \mathrm{m}^3\).

For this exercise:

• Given volume = \(350 \, \mathrm{mL}\),
• Converted volume = \(350 \times 10^{-6} \, \mathrm{m}^3 = 0.00035 \, \mathrm{m}^3\).

This conversion ensures accuracy when using the Ideal Gas Law, as it demands consistent units for proper calculation.

Propane Molar Mass

The molar mass of propane \(\mathrm{C}_3\mathrm{H}_8\) is an important factor in the Ideal Gas Law, as it helps in determining the number of moles \(n\).

Propane's molar mass is \(44.1 \, \mathrm{g/mol}\). To calculate moles:

• Given mass of propane = \(3.2 \, \mathrm{g}\),
• Number of moles \(n = \frac{3.2}{44.1} \approx 0.0725 \, \mathrm{mol}\).

Understanding the molar mass allows for the conversion from grams to moles, making it possible to use the gas law formula correctly.

Temperature Conversion

Temperature often needs to be converted from Celsius to Kelvin when using the Ideal Gas Law.

The conversion formula is:

• Kelvin \(K = \text{Celsius} + 273.15\).

For this problem:

• Given temperature = \(20^{\circ} \mathrm{C}\),
• Converted temperature = \(20 + 273.15 = 293.15 \, \mathrm{K}\).

Converting temperature to Kelvin ensures the correct application of the Ideal Gas Law, as Kelvin is the required unit of temperature in thermodynamic calculations.