Question

Assuming ideal behavior, calculate the density of gaseous HF at its normal boiling point \(\left(19.5^{\circ} \mathrm{C}\right)\). The experimentally measured density under the same conditions is \(3.10 \mathrm{~g} / \mathrm{L}\). Account for the discrepancy between your calculated value and the experimental result.

Short answer

Ideal density is 0.833 g/L; experimental density is 3.10 g/L, due to HF molecule associations in gas phase.

Step-by-step solution

Convert temperature to Kelvin

Since we are given the boiling point in degrees Celsius, we need to convert it to Kelvin using the formula \( T(K) = T(°C) + 273.15 \). For HF, the boiling point is \( 19.5^{\circ}C \), so \( T(K) = 19.5 + 273.15 = 292.65 \) K.

Use the Ideal Gas Law to find molar volume

The Ideal Gas Law is given by \( PV = nRT \). The molar volume \( V_m \) at standard conditions (1 atm, 292.65 K) using the ideal gas constant \( R = 0.0821 \, \text{L atm mol}^{-1} \text{K}^{-1} \) is found by rearranging to \( V_m = \frac{RT}{P} \). This becomes \( V_m = \frac{0.0821 \times 292.65}{1} \approx 24.03 \, \text{L mol}^{-1} \).

Calculate the molar mass of HF

The molar mass of HF is calculated by adding the atomic masses of hydrogen (1.01 g/mol) and fluorine (19.00 g/mol), which results in \( M = 1.01 + 19.00 = 20.01 \) g/mol.

Calculate theoretical density using molar volume and molar mass

Density \( \rho \) can be calculated by dividing molar mass \( M \) by molar volume \( V_m \), i.e., \( \rho = \frac{M}{V_m} = \frac{20.01}{24.03} \approx 0.833 \, \text{g/L} \).

Compare calculated and experimental densities

The calculated density is \( 0.833 \, \text{g/L} \) while the experimental density is \( 3.10 \, \text{g/L} \). The significant discrepancy can be attributed to the association of HF molecules in the gas phase, such as hydrogen bonding in clusters, which increases the apparent molar mass, thus increasing density.

Key concepts

Density Calculation

Calculating the density of a gas involves combining several principles from chemistry. Density is essentially how much mass is contained in a given volume; mathematically, \[ \rho = \frac{m}{V} \]where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume. In the context of gases, especially those behaving ideally, this concept is explored using the Ideal Gas Law. The steps usually involve calculating the molar mass of the gas, which is the mass of one mole of the gas. Once you know the molar mass, it can be divided by the volume of one mole of the gas at those conditions (the molar volume) to get the density. This process gives a theoretical density, which can then be compared to an experimental value. By comparing the two, additional factors such as molecular interactions unseen in ideal behavior can be inferred, providing insight into the nature of the gas.

Gaseous HF

Hydrogen fluoride (HF) in its gaseous form is a simple diatomic molecule. However, it behaves uniquely compared to many other gases due to its tendency to engage in strong hydrogen bonding. In the gaseous state, HF is not just a collection of isolated molecules. Instead, the molecules often form pairs or clusters due to the attractive forces between the hydrogen atom of one molecule and the fluorine atom of another. This can significantly alter its behavior and properties compared to what is predicted by the Ideal Gas Law. The discrepancy between calculated and experimental densities of HF can often be attributed to these non-ideal interactions that increase the effective molar mass and, consequently, the density.

Molar Volume

The molar volume of a gas refers to the volume that one mole of the gas occupies under specific conditions of temperature and pressure. For an ideal gas, the molar volume can be calculated using the Ideal Gas Law formula: \[ V_m = \frac{RT}{P} \]where \( R \) is the ideal gas constant, \( T \) the temperature in Kelvin, and \( P \) the pressure. Standard conditions for gases are often at 1 atm pressure, but temperature can vary, as in this HF example where the boiling point temperature is used. For gases like HF that do not behave ideally, the experimentally determined molar volumes can differ, primarily due to intermolecular forces not accounted for in the Ideal Gas Law. These differences help chemists understand deviations from ideal behavior.

Hydrogen Bonding

Hydrogen bonding is a strong type of dipole-dipole attraction that occurs between molecules when hydrogen is directly bonded to a small, highly electronegative atom like fluorine, oxygen, or nitrogen. HF is a prime example of a molecule that exhibits hydrogen bonding due to the considerable electronegativity of the fluorine atom. In HF, the hydrogen bonds lead to a higher level of association among molecules even in the gas phase. This means the gas does not exist solely as free molecules but forms clusters. These clusters make HF behave like a heavier species, which affects its physical properties such as density. This phenomenon explains why the measured density of gaseous HF is significantly higher than the density calculated assuming ideal gas behavior. Understanding hydrogen bonding is thus essential for interpreting experimental results in gas phase chemistry.