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 of the ethanol vapor is 5.67 atm.

Step 1: Determine Molar Mass

To find the pressure, we first need the molar mass of ethanol, \( \text{C}_2\text{H}_5\text{OH} \). Calculate it by adding the atomic masses: \( 2 \times \text{C} (12 \text{ g/mol}) + 6 \times \text{H} (1 \text{ g/mol}) + 1 \times \text{O} (16 \text{ g/mol}) = 46 \text{ g/mol} \).

Step 2: Convert Mass to Moles

Use the molar mass to convert the given mass of ethanol to moles. Use the formula: \( n = \frac{\text{mass}}{\text{molar mass}} \). This gives \( n = \frac{1.50 \text{ g}}{46 \text{ g/mol}} = 0.0326 \text{ mol} \).

Step 3: Convert Volume to Liters

Since 1 liter equals 1000 cm³, convert the volume of the cylinder from cm³ to liters: \( V = \frac{251 \text{ cm}^3}{1000} = 0.251 \text{ liters} \).

Step 4: Convert Temperature to Kelvin

To use the Ideal Gas Law, convert the temperature from Celsius to Kelvin: \( T = 250^\circ \text{C} + 273.15 = 523.15 \text{ K} \).

Step 5: Apply the Ideal Gas Law

Use the Ideal Gas Law formula \( PV = nRT \) to find the pressure (\( P \)). Rearrange to \( P = \frac{nRT}{V} \). Where \( R = 0.0821 \text{ L atm/mol K} \). Substitute the values: \( P = \frac{0.0326 \text{ mol} \times 0.0821 \text{ L atm/mol K} \times 523.15 \text{ K}}{0.251 \text{ L}} = 5.67 \text{ atm} \).

Key Concepts

Molar Mass

The molar mass of a substance is the mass of one mole of its molecules. It is essential for converting mass into moles, which is necessary for using the Ideal Gas Law. For ethanol, denoted as \( \text{C}_2\text{H}_5\text{OH} \), we calculate the molar mass by adding the atomic masses of its constituent atoms:

• Carbon (C): 2 atoms, each 12 g/mol = 24 g/mol
• Hydrogen (H): 6 atoms, each 1 g/mol = 6 g/mol
• Oxygen (O): 1 atom, 16 g/mol = 16 g/mol

Add these together to get the molar mass: \( 24 + 6 + 16 = 46 \text{ g/mol}\). This figure tells you that one mole of ethanol weighs 46 grams. With this information, you can convert any given mass of ethanol into the number of moles present, using the formula \( n = \frac{\text{mass}}{\text{molar mass}} \). This is pivotal in determining quantities needed in chemical reactions.

Temperature Conversion

Temperature conversion is critical in gas law calculations since gas laws demand temperature in Kelvin, not Celsius or Fahrenheit. The Kelvin scale is an absolute temperature scale starting at absolute zero, where all motion stops.
Converting from Celsius to Kelvin is straightforward. You add 273.15 to the Celsius temperature. For example, the temperature given in the problem is \( 250^\circ \text{C} \). Converting it to Kelvin involves simple addition: \( 250 + 273.15 = 523.15 \text{ K} \).
Temperatures in Kelvin facilitate gas law calculations because the scale is directly proportional to the motion of particles in a gas. This means that doubling the Kelvin temperature doubles the kinetic energy of the gas particles. Hence, always ensure your temperatures are in Kelvin when using formulas like the Ideal Gas Law.

Volume Conversion

Volume conversion is necessary when using gas law formulas, which typically require volumes to be in liters. The problem states a volume in cubic centimeters (cm³), but you need to convert this to liters by knowing the conversion factor.

• 1 liter = 1000 cubic centimeters

Given a volume of 251 cm³, divide by 1000 to convert:\[V = \frac{251 \text{ cm}^3}{1000} = 0.251 \text{ liters}\]By converting to liters, you're ensuring consistency with other units typically used in the Ideal Gas Law equation. This conversion is crucial for calculations involving any type of gas behavior where volumes are relevant.

Pressure Calculation

Pressure calculation ties everything together using the Ideal Gas Law \( PV = nRT \). This law relates an ideal gas's pressure, volume, temperature, and number of moles.
Before calculating pressure, all other units must be correctly set: convert grams to moles, Celsius to Kelvin, and cm³ to liters. Then, you substitute these values into the Ideal Gas Law.

• \( n \): moles of ethanol \( = 0.0326 \text{ mol} \)
• \( R \): ideal gas constant \( = 0.0821 \text{ L atm/mol K} \)
• \( T \): temperature in Kelvin \( = 523.15 \text{ K} \)
• \( V \): volume in liters \( = 0.251 \text{ L} \)

Rearrange the formula to solve for pressure \( P \):\[P = \frac{nRT}{V} = \frac{0.0326 \text{ mol} \times 0.0821 \text{ L atm/mol K} \times 523.15 \text{ K}}{0.251 \text{ L}} = 5.67 \text{ atm}\]This calculation shows that the ethanol vapor inside the steel cylinder exerts a pressure of 5.67 atm at the given conditions. Understanding this allows you to predict the gas's behavior under different conditions.