Problem 20 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Calculate the normal boling point of ethanol given that \(\Delta_{\text {vap }} H=+42.6 \mathrm{kJ} \mathrm{mol}^{-1}\) and \(\Delta_{\text {vap }} \mathrm{S}=+122.0 \mathrm{JK}^{-1} \mathrm{mol}^{-1}\) (Section \(14.5)\)

Step-by-Step Solution

Answer

The boiling point of ethanol is approximately 76.03 °C.

Step 1: Understanding the Given Values

We are given the values of enthalpy change of vaporization (\(\Delta_{\text{vap}} H\)) and entropy change of vaporization (\(\Delta_{\text{vap}} S\)) for ethanol. These values are:\(\Delta_{\text{vap}} H = +42.6 \, \text{kJ} \text{mol}^{-1}\) \(\Delta_{\text{vap}} S = +122.0 \, \text{J K}^{-1} \text{mol}^{-1}\). Note that we need to convert \(\Delta_{\text{vap}} H\) to \(\text{J} \text{mol}^{-1}\) by multiplying by 1000: \(\Delta_{\text{vap}} H = +42600 \, \text{J} \text{mol}^{-1}\).

Step 2: Recalling the Relation Between Thermodynamic Quantities

At the boiling point, the change in Gibbs free energy (\(\Delta G\)) is zero. The equation describing this is \(\Delta G = \Delta H - T\Delta S = 0\). From this equation, we can find the boiling point \(T\) by rearranging to \(T = \frac{\Delta H}{\Delta S}\).

Step 3: Calculating the Boiling Point

Substitute the given values into the formula \(T = \frac{\Delta H}{\Delta S}\):\[ T = \frac{42600 \, \text{J mol}^{-1}}{122.0 \, \text{J K}^{-1} \text{mol}^{-1}} \]Perform the division to get:\[ T \approx 349.18 \, \text{K} \]

Step 4: Converting the Temperature to Celsius

Convert the calculated temperature from Kelvin to Celsius using the relation \(T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15\):\[ T_{\text{Celsius}} = 349.18 - 273.15 = 76.03 \, \text{°C} \]

Key Concepts

Enthalpy

Enthalpy is a fundamental concept in thermodynamics representing the total heat content of a system. It is denoted by the symbol \( H \). When we talk about changes in enthalpy, such as during phase transitions like boiling, we refer to it as \( \Delta H \). The enthalpy of vaporization (\( \Delta_{\text{vap}} H \)) represents the energy required to change a substance from a liquid to a gas at a constant pressure. In the context of ethanol, \( \Delta_{\text{vap}} H \) was given as \( +42.6 \, \text{kJ} \, \text{mol}^{-1} \), which was converted to \( +42600 \, \text{J} \, \text{mol}^{-1} \) to match the units with entropy for calculation purposes. This energy input is significant as it determines how much heat is required for ethanol to overcome molecular interactions and enter the gaseous state.

Entropy

Entropy, denoted by \( S \), is a measure of randomness or disorder in a system. In the realm of thermodynamics, entropy offers insight into the energy dispersal within a system. With phase changes, like the transition of a liquid to a gas, entropy changes significantly. For ethanol, the entropy change of vaporization (\( \Delta_{\text{vap}} S \)) is \( +122.0 \, \text{J K}^{-1} \, \text{mol}^{-1} \). This positive value indicates an increase in disorder as ethanol molecules move from the ordered liquid phase to the more disordered gaseous phase.
Understanding entropy helps predict the spontaneity of processes and temperature dependencies, making it crucial for calculating conditions like boiling points.

Gibbs Free Energy

Gibbs Free Energy, symbolized by \( G \), is a thermodynamic potential that measures the usable energy or work potential in a system at constant temperature and pressure. It is defined by the equation: \[ G = H - TS \]where \( H \) is enthalpy, \( T \) is temperature, and \( S \) is entropy. A vital point to note is that at the boiling point, the change in Gibbs Free Energy (\( \Delta G \)) becomes zero. This is because a phase change like boiling occurs when a system is at equilibrium. Using the equation for \( \Delta G \): \[ \Delta G = \Delta H - T \Delta S = 0 \]We rearrange it to \( T = \frac{\Delta H}{\Delta S} \), which allows us to calculate the boiling point of substances such as ethanol. Understanding \( \Delta G \) is key to predicting reactions and phase changes under given conditions.

Boiling Point Calculation

Boiling point calculation involves determining the temperature at which a liquid becomes a gas, driven by overcoming intermolecular forces. To find this point for ethanol, it's essential to use the equation involving Gibbs Free Energy at equilibrium: \[ T = \frac{\Delta H}{\Delta S} \]Given the provided values for ethanol:

• \( \Delta_{\text{vap}} H = 42600 \, \text{J mol}^{-1} \)
• \( \Delta_{\text{vap}} S = 122.0 \, \text{J K}^{-1} \text{mol}^{-1} \)

Substitute these values to find the boiling point:\[ T = \frac{42600}{122.0} \approx 349.18 \, \text{K} \]Once calculated, convert to Celsius by subtracting 273.15:\[ T_{\text{Celsius}} = 349.18 - 273.15 = 76.03 \, \text{°C} \]This calculation highlights the practical application of thermodynamic principles to predict real-world scenarios.