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.)

Short answer

The pressure of the ethanol vapor is approximately 5.56 atm.

Step-by-step solution

Convert mass to moles

Find the molar mass of ethanol, \(\text{C}_2\text{H}_5\text{OH}\). The molar mass is calculated as \((2 \times 12.01) + (6 \times 1.01) + 16.00 = 46.08\, \text{g/mol}\). Given \(1.50\, \text{g}\) of ethanol, calculate the number of moles, using the formula: \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\). Thus, \(\text{moles} = \frac{1.50}{46.08} \approx 0.0325\, \text{mol}\).

Convert temperature to Kelvins

Convert the temperature from Celsius to Kelvin using the formula: \(T(K) = T(\degree C) + 273.15\). So, \(250^{\circ} \text{C} = 250 + 273.15 = 523.15\, \text{K}\).

Convert volume to liters

Convert the volume from cubic centimeters to liters. Since \(1\, \text{L} = 1000\, \text{cm}^3\), we have \(V = 251\, \text{cm}^3 \times \frac{1\, \text{L}}{1000\, \text{cm}^3} = 0.251\, \text{L}\).

Use the ideal gas law

Use the ideal gas law to find the pressure. The ideal gas law is given by \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is moles, \(R\) is the ideal gas constant \(0.0821\, \text{L} \cdot \text{atm} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}\), and \(T\) is temperature in Kelvin. Rearranging for \(P\), we get \(P = \frac{nRT}{V}\).

Calculate pressure

Substitute the known values into the equation: \(P = \frac{0.0325 \times 0.0821 \times 523.15}{0.251}\). This simplifies to \(P \approx \frac{1.396}{0.251} \approx 5.56\, \text{atm}\).

Key concepts

Ethanol Vapor Pressure

Understanding ethanol vapor pressure is essential when studying its behavior in different conditions. Ethanol is a colorless liquid that can evaporate very quickly, creating vapor—an essential property in chemistry.

Ethanol vapor pressure refers to the pressure exerted by the ethanol molecules in the vapor phase above the liquid. It's important because it helps determine how easily ethanol will evaporate at a given temperature. As temperature increases, so does the kinetic energy of ethanol molecules, raising the vapor pressure.

• At higher temperatures, ethanol has higher vapor pressures, meaning it evaporates more quickly.
• Vapor pressure is a key factor in applications like distillation and the food industry.

In the original exercise, understanding ethanol vapor pressure helps us use the ideal gas law to calculate the pressure it creates inside a container, aiding in essential calculations for chemical processes.

Molar Mass Calculation

Calculating the molar mass of a compound is a foundational skill in chemistry. The molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol).

For ethanol (C_2H_5OH), the molar mass is calculated by summing the atomic masses of all atoms in the molecule:

• Carbon (C): 2 atoms × 12.01 g/mol = 24.02 g/mol
• Hydrogen (H): 6 atoms × 1.01 g/mol = 6.06 g/mol
• Oxygen (O): 1 atom × 16.00 g/mol = 16.00 g/mol

Adding these together, ethanol's molar mass is 46.08 g/mol. This value allows us to convert the mass of ethanol into moles using the formula: \( ext{moles} = rac{ ext{mass}}{ ext{molar mass}}\). It's a crucial step in applying the ideal gas law, as it helps calculate the amount of substance present.

Temperature Conversion to Kelvin

Temperature conversion is an important aspect when dealing with gas laws and chemistry experiments. The Kelvin scale is the SI unit for temperature and provides an absolute reference point.

Converting Celsius to Kelvin is straightforward:

• Add 273.15 to the Celsius temperature.

For the exercise given:

• \(250^{ ext{°C}} + 273.15 = 523.15 ext{K}\),

Converting to Kelvin is necessary because the ideal gas law, \(PV = nRT\), requires the temperature to be in Kelvin to ensure all calculations are consistent. Kelvin provides a direct measure of kinetic energy measures—key to understanding how temperature affects gas behavior.

Volume Conversion to Liters

Volume conversion is another key element in chemical calculations. Many substances are measured in cubic centimeters (cm³) but chemists often need this in liters (L) for calculations using the ideal gas law.

To convert from cm³ to L:

• Remember that 1 L = 1000 cm³.
• Simply divide the volume in cm³ by 1000.

In the given exercise:

• 251 cm³ becomes \( 0.251 ext{ L}\) after dividing by 1000.

This conversion aligns unit measurements needed for gas law calculations, ensuring consistency and accuracy when using the equation \(PV = nRT\). Proper conversion prevents errors and facilitates smooth computations in a scientific context.