Problem 73 - Chemistry The Molecular Nature of Matter

Question

To three significant figures, calculate the density in \(\mathrm{g} \mathrm{L}^{-1}\) of the following gases at STP: (a) \(\mathrm{C}_{2} \mathrm{H}_{6}\) (ethane), (b) \(\mathrm{N}_{2}\), (c) \(\mathrm{Cl}_{2}\) (d) Ar.

Step-by-Step Solution

Answer

Density at STP: (a) \(C_2H_6\) = 1.36 \(\mathrm{g}\mathrm{L}^{-1}\), (b) \(N_2\) = 1.25 \(\mathrm{g}\mathrm{L}^{-1}\), (c) \(Cl_2\) = 3.21 \(\mathrm{g}\mathrm{L}^{-1}\), (d) Ar = 1.78 \(\mathrm{g}\mathrm{L}^{-1}\)

Step 1: Determine molar mass

Calculate the molar mass of each gas using the periodic table. Multiply the atomic mass by the number of each type of atom in the molecule.

Step 2: Use the molar volume at STP

At standard temperature and pressure (STP), 1 mole of any gas occupies 22.4 L. Use this volume to calculate density.

Step 3: Calculate the density

Divide the molar mass of each gas by the molar volume at STP to find the density in \(\mathrm{g}\mathrm{L}^{-1}\).

Step 4: Round to three significant figures

Round the calculated density to three significant figures, as required by the problem.

Key Concepts

Molar Mass

Determining the molar mass of a substance is a fundamental step in chemistry that allows us to make sense of the mass-to-substance ratio. Simply put, molar mass is the weight of one mole of a compound or element. You'll often find it expressed in grams per mole (g/mol). Remember, a mole is Avogadro's number (\(6.022 \times 10^{23}\) entities) of anything, which is why it's such a useful standard in chemistry.

When you calculate the molar mass, you look at the periodic table to find the atomic mass of each element. For a compound, you multiply the atomic mass of each element by the number of times it appears in a molecule and then add these values together. For instance, the molar mass of water ((O\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right).. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ((( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (( ((( ( ( (( (( ((( (((.Scheme.Scheme.Scheme.Scheme.Scheme.Scheme.Scheme.Scheme.( ( ( ((( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (((((((((((((((((((((((((((((((((((((((((((((((((( ( (( ((.( ( (recognizing the crucial role that molar_mass plays in calculating other properties, like density.

Standard Temperature and Pressure

When working with gases, scientists needed a common playing field to standardize their measurements. This is where Standard Temperature and Pressure, or STP, comes into play. At STP, the temperature is set at 0°C (273.15 K) and the pressure at 1 atmosphere (atm). These conditions give us a predictable point of reference, particularly handy for comparison and several calculations.

At STP, any gas has a molar volume — the volume one mole of gas occupies — of approximately 22.4 liters. This isn't just an arbitrary number; it's based on the ideal gas law. Why is the molar volume at STP such a powerful tool? It allows you to determine the density of a gas by applying a simple formula: Density = Molar mass / Molar volume. By using this standard, you can swiftly compare the densities of different gases under the same conditions, eliminating variables that might affect your results.

Significant Figures

Accuracy and precision are cornerstones of scientific measurements, and this is where significant figures come in. Significant figures, or 'sig figs', represent the digits in a number that contribute to its precision. They begin with the first non-zero digit and include all the digits after it, even the zeros that may serve as placeholders or reflect precise measurements.

When you round numbers to the appropriate number of significant figures, you are effectively communicating how exact your measurements or calculations are. Your solution's reliability hinges on these figures. For instance, if you’re told to round to three significant figures, a calculated density of 1.23456 g/L would be reported as 1.23 g/L. This indicates that the first three digits are the reliable ones, and the rest are beyond the precision of the measurement. Grasping how to use significant figures correctly is essential in scientific calculations to ensure that you neither overstate nor understate the precision of your results.