Problem 112 - Chemistry: The Central Science

Question

Cyclopropane, a gas used with oxygen as a general anesthetic, is composed of \(85.7 \% \mathrm{C}\) and \(14.3 \% \mathrm{H}\) by mass. (a) If \(1.56 \mathrm{~g}\) of cyclopropane has a volume of \(1.00 \mathrm{~L}\) at \(0.984 \mathrm{~atm}\) and \(50.0^{\circ} \mathrm{C}\), what is the molecular formula of cyclopropane? (b) Judging from its molecular formula, would you expect cyclopropane to deviate more or less than Ar from ideal-gas behavior at moderately high pressures and room temperature? Explain.

Step-by-Step Solution

Answer

The molecular formula of cyclopropane is C_3H_6. It would be expected to deviate more than Ar from ideal-gas behavior at moderately high pressures and room temperature due to its larger and more complex molecular structure, resulting in stronger intermolecular forces and greater molecular volume.

Step 1: Find moles of C and H

Let's assume that we have a 100 g sample of cyclopropane. In this sample, the mass of carbon is 85.7 g and the mass of hydrogen is 14.3 g. To find the moles of C and H in the sample: Moles of C = \(\frac{\text{mass of C}}{\text{molar mass of C}} = \frac{85.7 g}{12.01 g/mol}\) Moles of C \(= 7.14 mol\) Moles of H = \(\frac{\text{mass of H}}{\text{molar mass of H}} = \frac{14.3 g}{1.01 g/mol}\) Moles of H \(= 14.16 mol\)

Step 2: Find empirical formula of cyclopropane

The empirical formula is the simplest whole-number ratio of C and H in the molecule, so we need to find the ratio of moles of C and H. Divide moles of C and H by their smallest value: \(\frac{7.14}{7.14} = 1\) and \(\frac{14.16}{7.14} = 1.98 \approx 2\) Thus, the empirical formula is C_1H_2 or simply CH_2.

Step 3: Find molecular mass of empirical formula

The molecular mass of the empirical formula CH_2 is: Mass of C \(+ 2×\) Mass of H \(= 12.01 g/mol + 2×1.01 g/mol\) Molecular mass of CH_2 \(= 14.03 g/mol\)

Step 4: Use Ideal Gas Law to find moles of cyclopropane

We are given the volume, pressure, and temperature of 1.56 g sample of cyclopropane. Given the Ideal Gas Law: PV = nRT, where P = 0.984 atm V = 1.00 L R = 0.0821 L atm/mol K (gas constant) T = 50.0 + 273.15 = 323.15 K Rearrange Ideal Gas Law to find n: n = \(\frac{PV}{RT} = \frac{(0.984)(1.00)}{(0.0821)(323.15)}\) n \(= 0.0373 mol\)

Step 5: Find molecular mass of cyclopropane

Molecular mass of cyclopropane = \(\frac{\text{mass}}{\text{moles}}\) Molecular mass of cyclopropane \(\frac{1.56 g}{0.0373 mol}\) Molecular mass of cyclopropane \(= 41.83 g/mol\)

Step 6: Find molecular formula of cyclopropane

To find the molecular formula, we need to find the whole number multiple of empirical formula mass that is equal to the molecular mass of cyclopropane. Whole number multiple = \(\frac{\text{molecular mass of cyclopropane}}{\text{molecular mass of empirical formula}}\) Whole number multiple \(= \frac{41.83 g/mol}{14.03 g/mol}\) Whole number multiple \(= 2.98 \approx 3\) Since the whole number multiple is approximately 3, the molecular formula of cyclopropane is 3×(CH_2) = C_3H_6.

Step 7: Deviation from ideal-gas behavior

Cyclopropane (C_3H_6) is a larger and more complex molecule than Ar (argon), which is a monatomic gas. Deviations from ideal-gas behavior are more likely for larger and more complex molecules, due to stronger intermolecular forces and greater molecular volume. Thus, cyclopropane would be expected to deviate more than Ar from ideal-gas behavior at moderately high pressures and room temperature.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that describes the behavior of an ideal gas. It is articulated as \(PV = nRT\), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume.
• \(n\) is the number of moles.
• \(R\) is the universal gas constant.
• \(T\) is the absolute temperature in Kelvin.

The ideal gas law assumes that gas molecules do not interact with each other and occupy no volume. This is a simplification, used to make calculations easier.
When working with real gases like cyclopropane, we should be aware that these assumptions may not always perfectly hold. This is especially true at high pressures or low temperatures, where gas molecules are closer together, making interactions more likely.

Empirical Formula

An empirical formula represents the simplest whole-number ratio of elements in a compound. To find it, the masses of each element are converted to moles, and then these mole values are simplified to the smallest whole number ratio.

For cyclopropane, if we assume a 100 g sample, we find 85.7 g of carbon and 14.3 g of hydrogen. After using the molar mass to convert these to moles, we get:

• Moles of C: \[\frac{85.7 \text{ g}}{12.01 \text{ g/mol}} = 7.14 \text{ mol}\]
• Moles of H: \[\frac{14.3 \text{ g}}{1.01 \text{ g/mol}} = 14.16 \text{ mol}\]

Finding the smallest whole-number ratio involves dividing each mole value by the smallest of the two, which gives the empirical formula as \(\text{CH}_2\).
This is not the complete molecular structure, but it is the starting point in determining the actual molecular formula.

Molecular Formula

The molecular formula shows the actual number of each type of atom in a molecule. To find it, the empirical formula is multiplied by a whole number. This number is found by dividing the molecular mass by the empirical formula mass.

For cyclopropane, we first calculate the mass of the empirical formula \(\text{CH}_2\) which is:

• \[12.01 \text{ g (C)} + 2 \times 1.01 \text{ g (H)} = 14.03 \text{ g/mol}\]

Then, using the sample data:

• The obtained molecular mass from the ideal gas calculation was \(41.83 \text{ g/mol}\).

This results in a multiplier of about 3 (since \(\frac{41.83}{14.03} \approx 3\)). The empirical formula is then adjusted accordingly, and we find the molecular formula to be \(\text{C}_3\text{H}_6\), which reflects the true composition of cyclopropane.

Deviation from Ideal Gas Behavior

Real gases do not always behave as ideal gases. The deviation is due to intermolecular forces and the volume occupied by gas molecules. Larger, more complex molecules, like cyclopropane, tend to deviate more than smaller and simpler ones such as Argon (Ar).

For a gas like cyclopropane, deviations occur due to:

• Stronger intermolecular forces that aren't accounted for in the ideal gas model.
• Significant molecular volume that becomes non-negligible especially in high pressure conditions.

At moderately high pressures and room temperature, cyclopropane's molecular interactions and size make it likely to deviate more from ideal conditions compared to monoatomic gases like Argon.