Question

One method of preparing elemental mercury involves roasting cinnabar (HgS) in quicklime \((\mathrm{CaO})\) at \(600 .{ }^{\circ} \mathrm{C}\) followed by condensation of the mercury vapor. Given the heat of vaporization of mercury \((296 \mathrm{~J} / \mathrm{g})\) and the vapor pressure of mercury at \(25.0^{\circ} \mathrm{C}\) \(\left(2.56 \times 10^{-3}\right.\) torr), what is the vapor pressure of the condensed mercury at \(300 .{ }^{\circ} \mathrm{C} ?\) How many atoms of mercury are present in the mercury vapor at \(300 .{ }^{\circ} \mathrm{C}\) if the reaction is conducted in a closed 15.0-L container?

Short answer

The vapor pressure of the condensed mercury at 300°C is \(3.006 \times 10^{-9}\) atm. There are approximately \(6.458 \times 10^{12}\) mercury atoms in the vapor at 300°C in a 15.0L container.

Step-by-step solution

Find the vapor pressure of mercury at 300°C with the Clausius-Clapeyron equation

We will use the Clausius-Clapeyron equation to find the vapor pressure at 300°C. The equation is: \( \ln \frac{P_2}{P_1} = - \frac{L}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \) Where \(P_1\) is the initial vapor pressure (given at 25°C), \(P_2\) is the vapor pressure at 300°C, \(L\) is the heat of vaporization, \(R\) is the ideal gas constant, \(T_1\) is the initial temperature (25°C, in Kelvin), and \(T_2\) is the temperature at 300°C (in Kelvin). We need to convert the temperature from Celsius to Kelvin: \(T_1 = 25°C+ 273.15 = 298.15 K\) \(T_2 = 300°C + 273.15 = 573.15 K\) We will also need to convert the given vapor pressure from torr to atm: \(P_1 = 2.56 \times 10^{-3} \frac{\text{torr}}{1} \times \frac{1 \text{atm}}{760 \text{ torr}} = 3.368 \times 10^{-6} \text{ atm}\) Now we can rearrange the Clausius-Clapeyron equation for P2 and plug in the values: \(P_2 = P_1 \times \exp \left[- \frac{L}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \right] \) \(P_2 = 3.368 \times 10^{-6} \text{ atm} \times \exp \left[ - \frac{296 \frac{\text{J}}{\text{g}} \times 1 \frac{\text{g}}{200.59 \text{u}} \times 1000 \frac{\text{J}}{\text{kJ}}}{8.314 \frac{\text{J}}{\text{mol K}}}\left(\frac{1}{573.15 \text{K}} - \frac{1}{298.15 \text{K}} \right) \right]\)

Calculate the vapor pressure at 300°C

With the values plugged in, we can now find the vapor pressure of the condensed mercury at 300°C: \(P_2 = 3.368 \times 10^{-6} \text{ atm} \times \exp \left[ -\frac{0.00446 \frac{\text{kJ}}{\text{u mol}}} {8.314 \frac{\text{J}}{\text{mol K}}} \left( \frac{1}{573.15 \text{K}} - \frac{1}{298.15 \text{K}} \right) \right] \) \(P_2 = 3.368 \times 10^{-6} \text{ atm} \times \exp(-7.027) \) \(P_2 = 3.368 \times 10^{-6} \text{ atm} \times 8.919 \times 10^{-4} \) \(P_2 = 3.006 \times 10^{-9} \text{ atm} \) The vapor pressure of the condensed mercury at 300°C is \(3.006 \times 10^{-9}\) atm.

Find the number of atoms of mercury in the vapor at 300°C

Now we will use the Ideal Gas Law to find the number of mercury atoms at 300°C in a 15.0L container. The Ideal Gas Law is: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. We know P, V, R, and T, so we can solve for n: n = \(\frac{PV}{RT}\) n = \(\frac{(3.006 \times 10^{-9} \text{ atm})(15.0 \text{ L})}{(0.0821 \frac{\text{L atm}}{\text{mol K}})(573.15 \text{ K})}\) n = 1.073 \times 10^{-11} \text{ mol} Now, we can find the number of atoms by multiplying the number of moles by Avogadro's number: number of atoms = n × Avogadro's number number of atoms = \(1.073 \times 10^{-11} \text{ mol} \times 6.022 \times 10^{23} \frac{\text{atoms}}{\text{mol}}\) number of atoms = \(6.458 \times 10^{12}\) Hg atoms There are approximately \(6.458 \times 10^{12}\) mercury atoms in the vapor at 300°C in a 15.0L container.

Key concepts

Understanding the Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is fundamental in thermodynamics and physical chemistry as it describes the phase transitions between two states of matter, such as between a liquid and a gas. In the context of determining the vapor pressure of a substance like mercury, this equation relates the change in vapor pressure with temperature, considering the heat of vaporization.

The Clausius-Clapeyron equation is generally expressed as:
\[ \ln \left(\frac{P_2}{P_1}\right) = - \frac{L}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]
where:\

• \(P_1\) and \(P_2\) are the initial and final vapor pressures at temperatures \(T_1\) and \(T_2\), respectively,
• \(L\) is the heat of vaporization of the substance,
• \(R\) is the ideal gas constant,
• \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.

This equation helps us understand how the vapor pressure of a liquid or solid changes as the temperature increases or decreases, which is vital for many industrial processes and scientific research.

The Heat of Vaporization Explained

Heat of vaporization, often denoted as \(L\), is the amount of energy required to turn a substance from its liquid phase into a vapor without a change in temperature. It's a type of latent heat that is usually measured in joules per gram (J/g) or kilojoules per mole (kJ/mol).

The concept of the heat of vaporization is integral when calculating changes in vapor pressure using the Clausius-Clapeyron equation. For instance, mercury has a specific heat of vaporization value, and this energy is necessary to overcome the intermolecular forces holding the liquid mercury together and transition it into a gaseous state.

The higher the heat of vaporization, the stronger the intermolecular forces in the liquid, and consequently, more energy will be required to vaporize it. This property is pivotal in understanding why different substances have different vapor pressures at the same temperature.

Ideal Gas Law Application

The Ideal Gas Law is a cornerstone of gas laws, combining Boyle's Law, Charles's Law, and Avogadro's Law. It's represented by \(PV = nRT\), where:\

• \(P\) stands for pressure,
• \(V\) 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 law is crucial for calculations involving gaseous systems, such as determining the number of gas particles or moles in a given volume at known temperature and pressure conditions. In the process of calculating the atomic presence of mercury vapor, the Ideal Gas Law provides a way to relate the physical quantities and allows for the determination of the amount of gas in moles.

The Role of Avogadro's Number

Avogadro's number, \(6.022 \times 10^{23}\), is the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. It's named after Amedeo Avogadro, an Italian scientist who contributed to the fundamental definition of the mole concept in chemistry.

In practical terms, this number allows chemists and physicists to convert between the macroscopic measurements that we can easily observe (like volume and pressure) and the microscopic scale of atoms and molecules, linking the molar quantity to a countable number of particles. Hence, after determining the number of moles of mercury vapor using the Ideal Gas Law, we multiply it by Avogadro's number to find out the exact number of mercury atoms present in the vapor at a specific temperature and volume.