Question

The following procedure is a simple though somewhat crude way to measure the molar mass of a gas. A liquid of mass \(0.0184 \mathrm{~g}\) is introduced into a syringe like the one shown here by injection through the rubber tip using a hypodermic needle. The syringe is then transferred to a temperature bath heated to \(45^{\circ} \mathrm{C},\) and the liquid vaporizes. The final volume of the vapor (measured by the outward movement of the plunger) is \(5.58 \mathrm{~mL},\) and the atmospheric pressure is \(760 \mathrm{mmHg}\). Given that the compound's empirical formula is \(\mathrm{CH}_{2}\), determine the molar mass of the compound.

Short answer

The calculated molar mass of the gas is approximately 85.89 g/mol.

Step-by-step solution

Convert Volume from mL to L

The volume of gas is provided in milliliters, and we need to work with liters for the Ideal Gas Law. Convert 5.58 mL to liters by dividing by 1000. \[V = \frac{5.58}{1000} = 0.00558 \text{ L}\]

Convert Temperature from Celsius to Kelvin

The temperature must be in Kelvin for the Ideal Gas Law calculation. Convert 45°C to Kelvin by adding 273.15.\[T = 45 + 273.15 = 318.15 \text{ K}\]

Use Ideal Gas Law to Find Moles

Apply the Ideal Gas Law, PV = nRT, to find the number of moles n. - P is the pressure in atm (convert 760 mmHg to atm by dividing by 760): \[P = \frac{760}{760} = 1 \text{ atm}\]- R is the gas constant (0.0821 L·atm/mol·K).Substitute the values into the Ideal Gas Law:\[1 \text{ atm} \times 0.00558 \text{ L} = n \times 0.0821 \times 318.15 \text{ K}\]Solving for n:\[n = \frac{1 \times 0.00558}{0.0821 \times 318.15} = 2.142 \times 10^{-4} \text{ mol}\]

Calculate Molar Mass

Molar mass (M) is calculated by dividing the mass of the gas by the number of moles. The mass of the liquid is given as 0.0184 g.\[M = \frac{0.0184 \text{ g}}{2.142 \times 10^{-4} \text{ mol}}\]Carrying out the division:\[M = 85.89 \text{ g/mol}\]

Compare with Empirical Formula Mass

The empirical formula is CH2, which has an empirical formula mass of:\[\left(12.01 + (2 \times 1.008)\right) = 14.026 \text{ g/mol}\]Compare the calculated molar mass with the empirical formula mass to check for consistency.The closest whole number ratio between the empirical mass and experimental molar mass will further confirm the molecular formula.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a powerful tool to understand the behavior of gases under different conditions. It is represented by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas in the container.
• \( V \) is the volume that the gas occupies.
• \( n \) is the number of moles of the gas.
• \( R \) is the gas constant, which has a value of 0.0821 L·atm/mol·K.
• \( T \) is the temperature in Kelvin.

By rearranging the formula, you can solve for the number of moles \( n \), which can be critical for determining the properties of a gas. For example, in our exercise, after converting various measurements to the correct units, we used the Ideal Gas Law to find the number of moles in the gas sample.

Empirical Formula

An empirical formula is the simplest version of a chemical formula that shows the ratio of atoms in a compound. In our case, the compound's empirical formula is \( \text{CH}_2 \). This formula indicates that for every carbon atom, there are two hydrogen atoms in the compound.
To determine the molar mass of the compound, we first calculate the empirical formula mass. Each element's atomic mass is used: carbon is 12.01 g/mol and hydrogen is 1.008 g/mol. Therefore, the empirical mass is calculated as:

• \( 12.01 + (2 \times 1.008) = 14.026 \text{ g/mol} \).

After obtaining the molar mass from experiments, comparing it to the empirical formula mass helps verify the molecular formula. If the experimental molar mass is a simple multiple of the empirical mass, it confirms the empirical formula ratio accurately represents the compound.

Gas Volume Conversion

When working with gases, it is crucial to use the correct units for volume. The Ideal Gas Law requires the use of liters (L) for volume. If the initial volume is provided in milliliters, as in our example, you convert it to liters by dividing by 1000.
For instance, the problem states the gas volume as 5.58 mL. Therefore, it is converted to liters as follows:

• \( V = \frac{5.58}{1000} = 0.00558 \text{ L} \).

This conversion ensures that all units are consistent with the gas constant \( R \), aiding in precise and accurate calculations using the Ideal Gas Law.

Temperature Conversion

Temperature conversion is another important step in calculations involving gases. The Ideal Gas Law formulates its calculations using the Kelvin scale.
Degrees Celsius must be converted to Kelvin by adding 273.15 to the Celsius measurement. In our exercise, the temperature given was 45°C. Hence, it was converted into Kelvin like this:

• \( T = 45 + 273.15 = 318.15 \text{ K} \).

Working in Kelvin allows the proportionality in the Ideal Gas Law to remain consistent and accurate, allowing us to proceed further with calculations like determining the number of moles.