Question

Chloroform, formerly used as an anesthetic and now believed to be a carcinogen, has a heat of vaporization \(\Delta H_{\text {vaporization }}=\) \(31.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\). The change, \(\mathrm{CHCl}_{3}(l) \longrightarrow \mathrm{CHCl}_{3}(g)\) has \(\Delta S^{\circ}=94.2 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\). At what temperature do we expect \(\mathrm{CHCl}_{3}\) to boil (i.e., at what temperature will liquid and vapor be in equilibrium at 1 atm pressure)?

Short answer

The boiling point of \( \text{CHCl}_3 \) is \( T = \frac{31400 \text{ J/mol}}{94.2 \text{ J/(mol*K)}} \) Kelvin.

Step-by-step solution

Understanding Gibbs Free Energy

The boiling point of a substance can be found by calculating the temperature at which its Gibbs Free Energy of vaporization (\( \triangle G_{vaporization} \) is zero. The Gibbs Free Energy is given by the formula \( \triangle G = \triangle H - T\triangle S \) where \( \triangle H \) is the enthalpy change, \( T \) is the temperature, and \( \triangle S \) is the entropy change. At the boiling point, \( \triangle G = 0 \) because the liquid and vapor are in equilibrium.

Setting Gibbs Free Energy to Zero

Knowing that at the boiling point \( \triangle G = 0 \) and given that \( \triangle H_{vaporization} = 31.4 \) kJ/mol and \( \triangle S^{\textdegree} = 94.2 \) J/(mol*K), we can set up the equation \( 0 = \triangle H - T\triangle S \) to solve for the temperature (T).

Solving for Temperature

We rearrange the equation to solve for T, yielding \( T = \frac{\triangle H}{\triangle S} \). Substituting the given values and converting \( \triangle H \) to J/mol for consistency, we use \( T = \frac{31400 \text{ J/mol}}{94.2 \text{ J/(mol*K)}} \).

Calculating the Boiling Point

Carrying out the division yields the boiling point T. It is important to ensure that units are consistent throughout the calculation - since we have converted \( \triangle H \) from kJ to J, we don't need to change the units of \( \triangle S \) for the temperature to be in Kelvin.

Key concepts

Gibbs Free Energy

Gibbs Free Energy, symbolized as \( G \), is a thermodynamic property that can predict the direction of chemical reactions and phase changes. At equilibrium, this value is zero, showing there is no net change occurring; the processes forward and reverse are happening at the same rate. For a substance to boil, its Gibbs Free Energy of vaporization must reach zero under constant pressure and temperature conditions, indicating that its liquid and gaseous phases coexist in equilibrium. To find the boiling point of chloroform, or any substance, we calculate the temperature at which \( G \) perfectly balances out to zero, which happens during the phase change from liquid to gas.

In terms of mathematical representation, \( G \) is calculated using the formula \( \Delta G = \Delta H - T\Delta S \), where \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy. By calculating the point where \( \Delta G \) equals zero, we can ascertain the exact temperature at which a substance will boil at atmospheric pressure.

Enthalpy Change (\(\Delta H\))

The enthalpy change \( \Delta H \) refers to the total heat content change within a system during a process, such as boiling. It is a crucial parameter in understanding the energy required for the vaporization phase transition. For chloroform, or any substance undergoing a phase change from liquid to vapor, the enthalpy change is actually referring to the heat of vaporization, which is the energy needed to overcome intermolecular forces and convert a liquid into its gaseous form.

The heat of vaporization is always a positive value because energy is absorbed during the endothermic process of vaporization. A higher heat of vaporization means the substance requires more thermal energy to transform from a liquid to a gas. The magnitude of \( \Delta H \) can influence the boiling point — higher values generally imply a higher boiling point, as more heat is required to achieve \( \Delta G = 0 \).

Entropy Change (\(\Delta S\))

Entropy, symbolized as \( S \) and measured in Joules per mole per Kelvin \( (J/mol\cdot K) \), is a measure of the disorder or randomness in a system. \( \Delta S \) or entropy change, occurs when a system goes from one state to another, such as from liquid to gas during the boiling process. With vaporization, the entropy increases since the gaseous state has more disorder than the liquid state.

The change in entropy is a key piece of the puzzle when using the Gibbs Free Energy equation to determine a substance's boiling point. A higher \( \Delta S \) suggests that the system's disorder is significantly increasing, which is typical of a phase change from a structured liquid to a more random gas. This increase in entropy helps drive the reaction forward, and its value is crucial for calculating the temperature at which the entropy gain balances the energy absorbed, thus dictating the boiling point.

Heat of Vaporization

The heat of vaporization is the amount of heat energy required to convert one mole of a liquid into vapor at constant temperature and pressure, and it is synonymous with the enthalpy change of vaporization \( \Delta H_{vaporization} \). This value is critical for understanding the boiling process because it quantifies the energy barrier that must be overcome for a substance to transition between phases.

In the case of chloroform, a heat of vaporization of 31.4 kJ/mol means that each mole of liquid chloroform needs 31.4 kJ of energy to vaporize. This property inherently affects boiling point calculations, as it serves as a key component in the Gibbs Free Energy equation for determining the point at which liquid and vapor phases are in equilibrium. The higher the heat of vaporization, the more heat is needed for boiling, hence the direct correlation with boiling points.

Phase Equilibrium

Phase equilibrium refers to the state at which multiple phases of a substance, such as liquid and vapor, coexist at a balance without any net change in phases over time. At the boiling point, the liquid and vapor phases of a substance like chloroform are in equilibrium at a specific combination of temperature and pressure. This is the temperature where the Gibbs Free Energy of the process of vaporization reaches zero, marking the point where the rate of evaporation equals the rate of condensation.

The concept of phase equilibrium is central to understanding boiling point calculation because it's the condition that defines the boiling point under a given pressure. At this point, the energetic and entropic properties of a substance are precisely balanced, symbolizing a key point in the understanding of physical chemistry and thermodynamic processes. Calculating the boiling point involves determining the temperature at which this equilibrium is established at 1 atm pressure, using the equations and principles we've outlined.