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 difference between your calculated result and the experimental value.

Short answer

Calculated density is 0.793 g/L; difference due to non-ideal gas behavior.

Step-by-step solution

Convert Temperature to Kelvin

The temperature given is at the normal boiling point of HF, which is 19.5°C. We need to convert this to Kelvin since gas laws use absolute temperature. The conversion is given by:\[ T_{\text{K}} = T_{\text{°C}} + 273.15 \]So for this problem:\[ T_{\text{K}} = 19.5 + 273.15 = 292.65 \text{ K} \]

Use the Ideal Gas Law to Find Density

The ideal gas law is given by \( PV = nRT \). We know the molar mass \( M \) of HF (19.0 g/mol), and we need density \( \rho \) in terms of grams per liter.From the ideal gas law:\[ n = \frac{m}{M} = \frac{PV}{RT} \]The density \( \rho = \frac{m}{V} \) can be re-arranged as:\[ \rho = \frac{PM}{RT} \]Using standard pressure \( P = 1 \, \text{atm} \), \( R = 0.0821 \, \text{L} \, \text{atm} / \text{mol} \, \text{K} \), and \( T = 292.65 \, \text{K} \), the density becomes:\[ \rho = \frac{(1 \, \text{atm})(19.0 \, \text{g/mol})}{(0.0821 \, \text{L} \, \text{atm} / \text{mol} \, \text{K})(292.65 \, \text{K})} \approx 0.793 \, \text{g/L} \]

Compare Calculated and Experimental Densities

The calculated density using the ideal gas law is approximately 0.793 g/L, while the experimentally measured density is 3.10 g/L. The discrepancy is likely due to non-ideal behavior of the gas; HF has hydrogen bonding which makes it deviate from ideal gas behavior, resulting in a higher density.

Key concepts

Gas Density Calculation

Calculating gas density is a useful way to understand how close the molecules in a gas are packed together. One common method to find the density of a gas is by using the Ideal Gas Law. This law connects the pressure, volume, temperature, and number of moles of a gas, making it a versatile tool for various gas calculations.
First, to use the Ideal Gas Law for density calculations, we begin with the equation: \[ PV = nRT \] where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.
From the equation above, we can derive the formula for density \( \rho \):
\[ \rho = \frac{m}{V} = \frac{PM}{RT} \]

• \( \rho \) is the density of the gas.
• \( P \) is the pressure you choose to use, often standard pressure.
• \( M \) is the molar mass of the gas.
• \( R \) is the ideal gas constant, \( 0.0821 \; \text{L atm/mol K} \).
• \( T \) is the temperature converted to Kelvin.

Applying this formula helps predict how much mass is within a given volume of gas, but it's important to remember that the assumptions of ideal gas behavior might not always match reality.

Temperature Conversion to Kelvin

The Kelvin scale is essential in chemistry, especially when working with gas laws. Unlike the Celsius scale, Kelvin starts at absolute zero, making it an absolute scale that provides consistency in calculations.
To convert any temperature from Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
This can be written as:\[ T_{\text{K}} = T_{\text{°C}} + 273.15 \] For example, if the temperature is \( 19.5^{\circ} \text{C} \), converting it to Kelvin will result in:\[ T_{\text{K}} = 19.5 + 273.15 = 292.65 \text{ K} \]

• Why use Kelvin? Kelvin is used because all gas law calculations require absolute temperatures to ensure accuracy.
• Importance: It prevents negative temperatures which do not make physical sense in these scenarios.

Using Kelvin ensures gas calculations are valid and consistent, ruling out potential errors in computations due to arbitrary scales.

Non-Ideal Gas Behavior

Most gases behave ideally under ordinary conditions, meaning they follow the Ideal Gas Law quite closely. However, certain gases can deviate from ideal behavior due to intermolecular forces and the volume of the molecules.
The discrepancies often arise when:

• The gas is at high pressures.
• The gas is at very low temperatures.
• Strong intermolecular forces are present.

For example, gaseous HF shows non-ideal behavior primarily due to hydrogen bonding among its molecules, which significantly influences its physical properties.
In the case of HF, this intermolecular attraction increases the density compared to what's predicted by the Ideal Gas Law.
This means that for HF, corrections or different models, such as the Van der Waals equation, may be needed to account for these interactions:\[ \left ( P + \frac{a}{V^2} \right )(V-b) = nRT \] Where \( a \) and \( b \) are constants specific for each gas which consider intermolecular forces and the volume of gas molecules respectively.
Understanding these deviations helps in more accurate modeling and prediction of gas behavior in real-world applications.