Problem 20 - Chemistry and Chemical Reactivity

Question

A steel cylinder holds \(1.50 \mathrm{g}\) of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\). What is the pressure of the ethanol vapor if the cylinder has a volume of \(251 \mathrm{cm}^{3}\) and the temperature is \(250^{\circ} \mathrm{C}\) ? (Assume all of the ethanol is in the vapor phase at this temperature. \()\)

Step-by-Step Solution

Answer

The pressure is approximately 5.55 atm.

Step 1: Determine the Molar Mass of Ethanol

The molecular formula of ethanol is \( C_2H_5OH \). Its molar mass is calculated as follows: \[(2 \times 12.01) + (6 \times 1.01) + 16.00 = 46.08 \text{ g/mol}.\]

Step 2: Calculate the Moles of Ethanol

Given the mass of ethanol is \( 1.50 \text{ g} \). Moles \( (n) \) are calculated using \( n = \frac{\text{mass}}{\text{molar mass}} \). Thus, \[ n = \frac{1.50}{46.08} \approx 0.0325 \text{ moles}.\]

Step 3: Convert Volume to Liters

The volume given is \( 251 \text{ cm}^3 \). To convert to liters, divide by 1000. \[ \text{Volume} = \frac{251}{1000} = 0.251 \text{ liters}.\]

Step 4: Convert Temperature to Kelvin

Temperature in Celsius must be converted to Kelvin using \( K = \text{C} + 273.15 \). Thus, \[ 250^\circ \text{C} = 250 + 273.15 = 523.15 \text{ K}.\]

Step 5: Apply the Ideal Gas Law to Find Pressure

Use the ideal gas law equation \( PV = nRT \), where \( R = 0.0821 \, \text{L atm/mol K} \). Solve for pressure \( P \): \[ P = \frac{nRT}{V}.\]Substitute the values: \( n = 0.0325 \), \( R = 0.0821 \), \( T = 523.15 \), \( V = 0.251 \).\[ P = \frac{0.0325 \times 0.0821 \times 523.15}{0.251} \approx 5.55 \, \text{atm}.\]

Key Concepts

Molar Mass of Ethanol

To understand the molar mass of ethanol, we need to look at its molecular formula, which is \( C_2H_5OH \). Each element in a compound has its own atomic mass, and the molar mass is the sum of these atomic masses based on the number of each atom in the molecule. For ethanol:

• Carbon (\( C \)) has an atomic mass of \( 12.01 \text{ g/mol} \), and ethanol has two carbon atoms, resulting in \( 2 \times 12.01 = 24.02 \text{ g/mol} \).
• Hydrogen (\( H \)) has an atomic mass of \( 1.01 \text{ g/mol} \). Since there are six hydrogen atoms in ethanol, this contributes \( 6 \times 1.01 = 6.06 \text{ g/mol} \).
• Oxygen (\( O \)) has an atomic mass of \( 16.00 \text{ g/mol} \).

When you add these contributions together, you get the total molar mass of ethanol: \( 24.02 + 6.06 + 16.00 = 46.08 \text{ g/mol} \). Understanding the molar mass is crucial for converting grams of a substance to moles, which helps in chemical calculations like pressure, gas volume, and reactions.

Conversion to Kelvin

In many scientific calculations, especially those involving gases, it's essential to work with temperatures in Kelvin. The Kelvin scale starts at absolute zero, the theoretically lowest possible temperature. To convert Celsius to Kelvin, use the formula:

• \( K = \text{C} + 273.15 \)

This conversion accounts for the zero point on the Kelvin scale being 273.15 units away from that on the Celsius scale. For example, in the exercise, the temperature of \( 250^{\circ} \text{C} \) becomes \( 250 + 273.15 = 523.15 \text{ K} \). Using Kelvin ensures consistency when applying laws like the Ideal Gas Law, where absolute temperature is needed for calculations of pressure, volume, or moles.

Moles Calculation

The concept of the mole is central to chemistry because it allows us to count atoms and molecules by weighing them. To calculate moles from a given mass, you use the formula:

• \( n = \frac{\text{mass}}{\text{molar mass}} \)

Here, \( n \) is the number of moles. In the exercise, we are given \( 1.50 \text{ g} \) of ethanol. Using ethanol's molar mass of \( 46.08 \text{ g/mol} \), we find:

• \( n = \frac{1.50}{46.08} \approx 0.0325 \text{ moles} \)

This conversion is fundamental because knowing the moles of a substance allows you to use the Ideal Gas Law, which relates pressure, volume, and temperature.

Vapor Pressure and Ideal Gas Law

The problem involves finding the vapor pressure of ethanol, which can be determined using the Ideal Gas Law. This law states:

• \( PV = nRT \)

Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant \( (0.0821 \text{ L atm/mol K}) \), and \( T \) is the temperature in Kelvin. The law assumes gases behave "ideally," meaning their particles do not attract or repel each other and move randomly. In the exercise:

• \( n = 0.0325 \text{ moles} \)
• \( V = 0.251 \text{ liters} \)
• \( T = 523.15 \text{ K} \)

Substituting these into the Ideal Gas Law gives the pressure:

• \( P = \frac{0.0325 \times 0.0821 \times 523.15}{0.251} \approx 5.55 \text{ atm} \)

This calculation helps illustrate how the amount of substance, the volume of the container, and the temperature all interact to produce a certain pressure in gases, making the Ideal Gas Law a critical tool for understanding vapor pressure.