Question

In Calculate the densities (in grams per liter) of the following gases at \(97^{\circ} \mathrm{C}\) and \(755 \mathrm{~mm} \mathrm{Hg}\) (a) hydrogen chloride (b) sulfur dioxide (c) butane \(\left(\mathrm{C}_{4} \mathrm{H}_{10}\right)\)

Short answer

Question: Calculate the densities of hydrogen chloride (HCl), sulfur dioxide (SO2), and butane (C4H10) at a temperature of 97°C and a pressure of 755 mmHg. Answer: The densities of the gases are as follows: (a) Hydrogen chloride (HCl): 1.238 g/L (b) Sulfur dioxide (SO2): 2.160 g/L (c) Butane (C4H10): 1.972 g/L

Step-by-step solution

Convert the temperature and pressure values

First, let's convert the given temperature from Celsius to Kelvin. Temperature in Kelvin (T) = Temperature in Celsius + 273.15 T = 97°C + 273.15 T = 370.15 K Next, we'll convert the given pressure from mmHg to atmospheres. Pressure in atmospheres (P) = Pressure in mmHg * (1 atm / 760 mmHg) P = 755 mmHg * (1 atm / 760 mmHg) P = 0.9934 atm

Calculate the molecular weight of each gas

(a) Hydrogen chloride (HCl) Molecular weight = (1 * atomic weight of H) + (1 * atomic weight of Cl) = (1 * 1.008) + (1 * 35.45) = 36.458 g/mol (b) Sulfur dioxide (SO2) Molecular weight = (1 * atomic weight of S) + (2 * atomic weight of O) = (1 * 32.07) + (2 * 16.00) = 64.07 g/mol (c) Butane (C4H10) Molecular weight = (4 * atomic weight of C) + (10 * atomic weight of H) = (4 * 12.01) + (10 * 1.008) = 58.122 g/mol

Calculate the gas densities

Now, we can use the ideal gas equation and molecular weights to calculate the densities of each gas. Gas density (ρ) = (P * molecular weight) / (R * T) Use R = 0.0821 L atm/mol K (a) Hydrogen chloride (HCl) density ρ = (0.9934 atm * 36.458 g/mol) / (0.0821 L atm/mol K * 370.15 K) ρ = 1.238 g/L (b) Sulfur dioxide (SO2) density ρ = (0.9934 atm * 64.07 g/mol) / (0.0821 L atm/mol K * 370.15 K) ρ = 2.160 g/L (c) Butane (C4H10) density ρ = (0.9934 atm * 58.122 g/mol) / (0.0821 L atm/mol K * 370.15 K) ρ = 1.972 g/L The gas densities are as follows: (a) Hydrogen chloride (HCl): 1.238 g/L (b) Sulfur dioxide (SO2): 2.160 g/L (c) Butane (C4H10): 1.972 g/L

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the behavior of an ideal gas under a set of conditions. It is expressed as \( PV = nRT \), where \( P \) represents the pressure of the gas, \( V \) is the volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

This equation allows us to relate the physical properties of a gas, assuming it behaves ideally, which means the gas particles have no volume and there are no intermolecular forces between them. Although no real gas behaves perfectly ideally, many gases at high temperature and low pressure approximate the ideal behavior quite well.

• The pressure \( P \) and volume \( V \) in the ideal gas law are directly proportional to the temperature \( T \) and amount of gas \( n \) in moles.
• The ideal gas constant \( R \) is a universal value that provides the necessary unit conversions within the equation. The value of \( R \) can vary depending on the units used for pressure, volume, and temperature.

When calculating gas densities using the ideal gas law, as done in the provided exercise, we manipulate the equation to solve for the density \( \rho \), which gives us \( \rho = \frac{P \times MW}{R \times T} \) where \( MW \) is the molecular weight of the gas in grams per mole.

Molecular Weight

Molecular weight, also known as molecular mass, is the sum of the atomic weights of all atoms in a molecule. It is measured in atomic mass units (amu) per molecule or grams per mole (g/mol). The atomic weight of each element is found on the periodic table and is based on the weighted average of all naturally occurring isotopes of that element.

In the context of calculating gas densities, knowing the molecular weight is crucial because it allows us to relate the mass of the gas to the number of moles present, which is a factor in the ideal gas law equation.

• To calculate the molecular weight of a compound, you multiply the atomic weight of each element by the number of atoms of that element in the molecule and then sum all these values.
• The molecular weight is essential when converting from moles to grams, which is a step in determining the density of a gas.

The provided exercise demonstrated this by calculating the molecular weights of hydrogen chloride, sulfur dioxide, and butane, which are then used alongside the ideal gas law to find the densities of the gases under specified conditions.

Temperature and Pressure Conversion

Temperature and pressure conversion is a preliminary step in many thermodynamic calculations, including gas density determination using the ideal gas law. For temperature, the Kelvin scale is used because it starts at absolute zero and is therefore requisite for all gas law equations.

To convert Celsius to Kelvin, we add 273.15 to the Celsius temperature. When working with pressure, various units can be used, such as atmospheres (atm), millimeters of mercury (mmHg), or pascals (Pa). These units can be converted to one another using conversion factors.

• For instance, 1 atm is equivalent to 760 mmHg or 101,325 Pa.
• In calculations involving the ideal gas law, it is essential to use consistent units for pressure to match the units of the gas constant \( R \).

The exercise example converts a pressure from mmHg to atm and a temperature from Celsius to Kelvin, ensuring consistency with the values required for using the ideal gas law and for the calculation of gas densities.