Problem 60 - Chemistry and Chemical Reactivity

Question

The enthalpy of vaporization of liquid diethyl ether, \(\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O},\) is \(26.0 \mathrm{kJ} / \mathrm{mol}\) at the boiling point of \(35.0^{\circ} \mathrm{C} .\) Calculate \(\Delta S^{\circ}\) for a vapor-to-liquid transformation at \(35.0^{\circ} \mathrm{C}.\)

Step-by-Step Solution

Answer

\(\Delta S^{\circ} \approx 84.39 \text{ J/mol} \cdot \text{K}\).

Step 1: Understanding the Relationship between Enthalpy and Entropy

The change in entropy (\(\Delta S^{\circ}\)) for a phase change at constant pressure can be calculated using the formula:\[\Delta S^{\circ} = \frac{\Delta H^{\circ}}{T}\]where \(\Delta H^{\circ}\) is the enthalpy change (in joules) and \(T\) is the absolute temperature in kelvins.

Step 2: Convert Temperature to Kelvin

Since the temperature is given in degrees Celsius, convert it to kelvin by adding 273.15:\[T(K) = 35.0 + 273.15 = 308.15\, \text{K}\]

Step 3: Convert Enthalpy from kJ to J

The enthalpy given is in kilojoules per mole, so convert it to joules per mole for consistency with entropy units:\[\Delta H^{\circ} = 26.0 \times 1000 = 26000\, \text{J/mol}\]

Step 4: Calculate the Entropy Change

Using the formula from Step 1, plug in the values for \(\Delta H^{\circ}\) and \(T\):\[\Delta S^{\circ} = \frac{26000\, \text{J/mol}}{308.15\, \text{K}} \approx 84.39\, \text{J/mol}\cdot \text{K}\]

Key Concepts

Enthalpy of Vaporization

The enthalpy of vaporization is the amount of energy needed to convert a liquid into a gas at a constant pressure. It is an essential concept for understanding the energy dynamics in phase changes. Enthalpy, in general, is a measure of the total heat content in a thermodynamic system. In the context of vaporization, it is particularly focused on the energy required to overcome intermolecular forces during the transition from liquid to vapor.

Key points to remember about enthalpy of vaporization:

• Measured in kilojoules per mole (kJ/mol), indicating the energy required per amount of substance.
• Higher enthalpy means stronger forces between molecules that need to be broken to change phase.
• It's a critical factor in calculating changes in entropy during phase transitions.

This value is also important in industrial applications where phase changes are involved, such as in distillation or cooling processes.

In our example, the enthalpy of vaporization for diethyl ether is given as 26.0 kJ/mol at its boiling point. Understanding and applying this value is crucial for any calculations related to energy changes in the system.

Temperature Conversion to Kelvin

To perform thermodynamic calculations accurately, it's crucial to convert temperature measurements from Celsius to Kelvin. The Kelvin scale is an absolute temperature scale, where 0 K is the point at which all thermal motion ceases, known as absolute zero.

To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. This conversion aligns with the absolute scale required for entropy calculations because Kelvin degrees have the same size as Celsius degrees, providing a consistent measure for thermodynamic equations.

Steps to convert:

• Identify the temperature in Celsius (e.g., 35.0°C).
• Add 273.15 to this value to convert to Kelvin.

For our exercise, the temperature of 35.0°C converts to 308.15 K. This step ensures the accuracy of subsequent calculations of entropy.

Unit Conversion for Enthalpy

In thermodynamic calculations, consistency in units is vital. Enthalpy is often measured in kilojoules per mole (kJ/mol), but when calculating entropy, it's generally necessary to convert this into joules per mole (J/mol).

Converting from kJ to J entails a simple multiplication, as there are 1000 joules in a kilojoule. This conversion ensures that calculations involving joules, such as entropy changes in the formula \(\Delta S^{\circ} = \frac{\Delta H^{\circ}}{T}\), are correctly performed.

Steps to convert energy units:

• Take the given enthalpy value in kJ/mol (e.g., 26.0 kJ/mol).
• Multiply by 1000 to convert to J/mol (26000 J/mol in our example).

This process is straightforward yet critical to ensure calculations adhere to the unit requirements of the physical properties involved.

Phase Change

Phase change refers to the transition of matter from one state to another. Common examples are melting, freezing, vaporization, and condensation. Each phase change involves enthalpy changes and requires energy to be either absorbed or released.

In our exercise, the focus is on the transformation from liquid to vapor and back to liquid. The significance of phase changes in thermodynamics lies in their association with specific enthalpy and entropy changes.

Considerations for phase changes:

• Energy is absorbed during vaporization, breaking intermolecular bonds.
• Energy is released during condensation, forming new bonds.
• Phase changes occur at particular temperatures and pressures, such as the boiling point.

In particular, these processes are isothermal transformations, meaning they occur at constant temperature, which is crucial when applying formulas to calculate entropy changes. Mastering the concept of phase change is key to understanding the energy transfers that accompany state transitions.