Question

An aerosol spray can with a volume of \(125 \mathrm{~mL}\) contains \(1.30 \mathrm{~g}\) of propane gas \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) as a propellant. (a) If the can is at \(25^{\circ} \mathrm{C}\), what is the pressure in the can? (b) What volume would the butane occupy at \(S T P ?\) (c) The can's label says that exposure to temperatures above \(50^{\circ} \mathrm{C}\) may cause the can to burst. What is the pressure in the can at this temperature?

Short answer

The pressure in the can at 25°C is found using the Ideal Gas Law equation rearranged for P and plugging in the calculated values of n, V, and T: \[P = \frac{(1.3\mathrm{~g}/44.097\mathrm{~g/mol})\times(8.314\mathrm{~J/[mol\cdot K]})\times(298.15\mathrm{~K})}{125\mathrm{~mL}}\] The volume of propane at STP is calculated using the Ideal Gas Law equation rearranged for V and plugging in the values for n, R, and T: \[V_{STP} = \frac{(1.3\mathrm{~g}/44.097\mathrm{~g/mol})\times(8.314\mathrm{~J/[mol\cdot K]}/101.325\mathrm{~J/[L\cdot atm]})\times(273.15\mathrm{~K})}{1\mathrm{~atm}}\] Lastly, the pressure in the can at 50°C is calculated using the Ideal Gas Law equation rearranged for P and plugging in the calculated values of n, V, and T': \[P' = \frac{(1.3\mathrm{~g}/44.097\mathrm{~g/mol})\times(8.314\mathrm{~J/[mol\cdot K]})\times(323.15\mathrm{~K})}{125\mathrm{~mL}}\]

Step-by-step solution

Calculate the number of moles of propane gas

First, we need to determine the number of moles of propane gas in the aerosol can. To do this, we will use the molar mass of propane gas (C3H8) which is approximately \(44.097 \mathrm{~g/mol}\). To find the number of moles, we divide the mass of the propane gas by its molar mass: \[n = \frac{1.3\mathrm{~g}}{44.097\mathrm{~g/mol}}\]

Determine the temperature in Kelvin

The Ideal Gas Law equation requires temperatures to be expressed in Kelvin. Therefore, we need to convert the given temperatures (25°C and 50°C) to Kelvin. To convert Celsius temperature to Kelvin, we use the following formula: \[T(K) = T(°C) + 273.15\] For 25°C: \[T = 25 + 273.15 = 298.15\mathrm{~K}\] And for 50°C: \[T' = 50 + 273.15 = 323.15\mathrm{~K}\]

Calculate the pressure in the can at 25°C

Now that we have the number of moles and temperature in Kelvin, we can plug these values into the Ideal Gas Law equation alongside the can's volume to find the pressure at 25°C. \(PV = nRT\) We will rearrange the equation for P: \[P = \frac{nRT}{V}\] Using the values for n, V, and T: \[P = \frac{(1.3\mathrm{~g}/44.097\mathrm{~g/mol})\times(8.314\mathrm{~J/[mol\cdot K]})\times(298.15\mathrm{~K})}{125\mathrm{~mL}}\]

Calculate the volume of propane at STP

Now, we will calculate the volume that the propane would occupy at STP (Standard Temperature and Pressure). The standard conditions for temperature and pressure are T = 273.15 K and P = 1 atm. Using the Ideal Gas Law, we will solve for the volume at STP: \[V_{STP} = \frac{nR_{STP}T_{STP}}{P_{STP}}\] Where \(R_{STP} = \frac{R}{101.325\mathrm{~J/[L\cdot atm]}}\) Now we plug in the values for n, R, and T: \[V_{STP} = \frac{(1.3\mathrm{~g}/44.097\mathrm{~g/mol})\times(8.314\mathrm{~J/[mol\cdot K]}/101.325\mathrm{~J/[L\cdot atm]})\times(273.15\mathrm{~K})}{1\mathrm{~atm}}\]

Calculate the pressure in the can at 50°C

Lastly, we will find the pressure in the aerosol can at an increased temperature of 50°C (323.15 K) using the Ideal Gas Law equation: \[P' = \frac{nRT'}{V}\] Using the calculated values of n, V, and T': \[P' = \frac{(1.3\mathrm{~g}/44.097\mathrm{~g/mol})\times(8.314\mathrm{~J/[mol\cdot K]})\times(323.15\mathrm{~K})}{125\mathrm{~mL}}\] Now the student can plug in the given values for each step and solve for the pressure at 25°C, the volume at STP, and the pressure at 50°C.

Key concepts

Moles of Gas

To calculate the number of moles of propane gas, you need to know the mass of the gas and its molar mass. Moles give us a way to express amounts of a substance using the number of particles. For propane \((C_3H_8)\), the molar mass is approximately \(44.097 \, \text{g/mol}\).

Here's the formula to find the number of moles:

• Divide the mass of the propane by its molar mass.

Given:
Mass of propane \(= 1.30 \, \text{g}\)
Molar mass \(= 44.097 \, \text{g/mol}\)
The number of moles \(n\) is calculated as: \[n = \frac{1.3}{44.097} \, \text{mol}\] This calculation tells you how much of the gas is present in terms of moles.

Conversion to Kelvin

When dealing with gas laws, temperature must be in Kelvin because it avoids negative numbers and reflects an absolute scale. Converting Celsius to Kelvin is straightforward.

Here's how you convert:

• Add 273.15 to the Celsius temperature.

For \(25^{\circ} \text{C}:\)
\[T = 25 + 273.15 = 298.15 \, \text{K}\] For \(50^{\circ} \text{C}:\)
\[T' = 50 + 273.15 = 323.15 \, \text{K}\]
Making sure your temperatures are in Kelvin ensures accurate calculations when using the Ideal Gas Law.

Standard Temperature and Pressure

Standard Temperature and Pressure (STP) is used as a reference in scientific calculations. It's defined as a temperature of \(273.15 \, \text{K}\) and pressure of \(1 \, \text{atm}\).

Using STP allows for consistent measurements and comparisons. At STP, a mole of any ideal gas occupies \(22.414 \, \text{L}\). This condition helps in understanding how gases behave in typical conditions. In problems, you often use STP to predict the behavior of gases when the temperature and pressure are standardized.

Pressure Calculation

Pressure calculation in gases can be done using the Ideal Gas Law: \(PV = nRT\). Here, \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin.

The rearranged formula for pressure is: \[P = \frac{nRT}{V}\] To find pressure at given conditions:

• Plug in values for \(n, R, T, \) and \(V\).
• Use \((8.314 \, \text{J/mol·K})\) for \(R\).
• Check units; volume should be in \(\text{liters}\).

For example, calculate the pressure at \(25^{\circ} \text{C}\):
\[P = \frac{\left(\frac{1.3}{44.097}\right) \times 8.314 \times 298.15}{0.125}\]
This method helps quickly find the pressure of a gas under specific conditions.