Problem 32
Question
Diethyl ether, \(\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O},\) was once used as an anesthetic. Calculate the entropy change, \(\Delta_{\mathrm{r}} S^{\circ},\) for the vaporization of ether if its heat of vaporization is \(26.0 \mathrm{~kJ} / \mathrm{mol}\) at the boiling point of \(35.0^{\circ} \mathrm{C}\).
Step-by-Step Solution
Verified Answer
The entropy change \(\Delta_{r} S^{\circ}\) for vaporization is 84.36 J/(mol K).
1Step 1: Convert Temperature to Kelvin
First, convert the given boiling point from Celsius to Kelvin using the formula: \( T(K) = T(^{\circ}C) + 273.15 \). Here, \( T = 35.0 + 273.15 = 308.15 \, K \).
2Step 2: Recall the Formula for Entropy Change
The formula to calculate the change in entropy for vaporization is given by: \( \Delta_{r} S^{\circ} = \frac{\Delta_{v} H^{\circ}}{T} \).
3Step 3: Plug in the Values
Substitute the values for the heat of vaporization and temperature in Kelvin into the formula. \[ \Delta_{r} S^{\circ} = \frac{26.0 \, \mathrm{kJ/mol}}{308.15 \, \mathrm{K}} \].
4Step 4: Calculate Entropy Change
Perform the division to calculate the entropy change. First, convert \( \Delta_v H^{\circ} \) to joules by noting \( 26.0 \, \mathrm{kJ/mol} = 26000 \, \mathrm{J/mol} \). Then, \( \Delta_{r} S^{\circ} = \frac{26000 \, \mathrm{J/mol}}{308.15 \, \mathrm{K}} = 84.36 \, \mathrm{J/(mol \cdot K)} \).
Key Concepts
VaporizationHeat of VaporizationThermodynamic Calculations
Vaporization
Vaporization is the process where a liquid changes into a gas. It's essential not just in everyday life but also in various industrial and scientific applications. Vaporization occurs when molecules within a liquid gain enough energy to overcome intermolecular forces and enter the gaseous phase. This transition requires energy, often supplied in the form of heat. This energy is used to break the bonds holding the liquid molecules close together.
Understanding vaporization helps us grasp why certain substances evaporate easily, like alcohol or ether, while others do not. The process is more efficient at higher temperatures as molecules have more kinetic energy.
We often categorize vaporization into two types:
Understanding vaporization helps us grasp why certain substances evaporate easily, like alcohol or ether, while others do not. The process is more efficient at higher temperatures as molecules have more kinetic energy.
We often categorize vaporization into two types:
- Evaporation: This occurs at the surface of a liquid at any temperature below boiling point.
- Boiling: This occurs throughout the liquid once it reaches its boiling point.
Heat of Vaporization
The heat of vaporization is the amount of energy required to transform a given quantity of a substance from liquid to gas.
This is a crucial concept in understanding energy exchanges during phase transitions. Often denoted as \( \Delta_{v} H \), it is typically expressed in kilojoules per mole (kJ/mol) and reflects a fundamental property of a substance.
For diethyl ether in the given exercise, the heat of vaporization is 26.0 kJ/mol. This means that each mole of ether requires 26 kJ of energy to change from liquid to a gaseous state at its boiling point. This can help in various calculations, like determining the efficiency of a refrigerant in a cooling cycle or understanding an anesthetic's behavior when it evaporates.
In practice, knowing the heat of vaporization helps align with safety standards when using volatile substances by predicting the environmental conditions that trigger a phase change.
This is a crucial concept in understanding energy exchanges during phase transitions. Often denoted as \( \Delta_{v} H \), it is typically expressed in kilojoules per mole (kJ/mol) and reflects a fundamental property of a substance.
For diethyl ether in the given exercise, the heat of vaporization is 26.0 kJ/mol. This means that each mole of ether requires 26 kJ of energy to change from liquid to a gaseous state at its boiling point. This can help in various calculations, like determining the efficiency of a refrigerant in a cooling cycle or understanding an anesthetic's behavior when it evaporates.
In practice, knowing the heat of vaporization helps align with safety standards when using volatile substances by predicting the environmental conditions that trigger a phase change.
Thermodynamic Calculations
Thermodynamic calculations help predict how energy is exchanged during a chemical or physical process. Specifically, we can calculate changes in properties such as entropy, enthalpy, and free energy.
In the context of vaporization, one crucial calculation is determining the entropy change, \( \Delta_{r} S^{\circ} \), which gives insight into the disorder or randomness associated with the phase transition. The formula used is \( \Delta_{r} S^{\circ} = \frac{\Delta_{v} H^{\circ}}{T} \), where \( T \) is the temperature in Kelvin. It quantifies the energy dispersion during vaporization, reflecting the system's increased randomness as molecules move from a structured liquid phase to a more random gaseous phase.
In our example, diethyl ether's calculation involves converting the given heat of vaporization into joules and substituting it into the entropy change equation. This conversion is necessary because the entropy formula uses the SI unit "J/(mol*K)" for \( \Delta_{r} S^{\circ} \). So, 26.0 kJ/mol becomes 26,000 J/mol. Dividing this value by the temperature 308.15 K yields an entropy change of 84.36 J/(mol*K).
Thermodynamic calculations like this provide essential insight into how different variables impact state changes, aiding in the design of systems like engines or refrigeration devices.
In the context of vaporization, one crucial calculation is determining the entropy change, \( \Delta_{r} S^{\circ} \), which gives insight into the disorder or randomness associated with the phase transition. The formula used is \( \Delta_{r} S^{\circ} = \frac{\Delta_{v} H^{\circ}}{T} \), where \( T \) is the temperature in Kelvin. It quantifies the energy dispersion during vaporization, reflecting the system's increased randomness as molecules move from a structured liquid phase to a more random gaseous phase.
In our example, diethyl ether's calculation involves converting the given heat of vaporization into joules and substituting it into the entropy change equation. This conversion is necessary because the entropy formula uses the SI unit "J/(mol*K)" for \( \Delta_{r} S^{\circ} \). So, 26.0 kJ/mol becomes 26,000 J/mol. Dividing this value by the temperature 308.15 K yields an entropy change of 84.36 J/(mol*K).
Thermodynamic calculations like this provide essential insight into how different variables impact state changes, aiding in the design of systems like engines or refrigeration devices.
Other exercises in this chapter
Problem 30
Without consulting a table of standard molar entropies, predict whether \(\Delta_{1} S_{\text {system }}^{\circ}\) is positive or negative for each of these rea
View solution Problem 31
Calculate the entropy change, \(\Delta_{\mathrm{r}} S^{\circ},\) for the vaporization of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},\) at the boiling
View solution Problem 33
Calculate \(\Delta_{\mathrm{r}} S^{\circ}\) for each substance when the quantity of thermal energy indicated is transferred reversibly to the system at the temp
View solution Problem 34
Calculate \(\Delta_{\mathrm{r}} S^{\circ}\) for each of these substances when the quantity of thermal energy indicated is transferred reversibly to the system a
View solution