Calculating Entropy Change For Water Vaporization A Comprehensive Guide

In the realm of thermodynamics, entropy change (ΔS) stands as a fundamental concept, quantifying the degree of disorder or randomness within a system. This article delves into the calculation of ΔS for the conversion of one mole of water to vapor at 100°C, a process of significant practical and theoretical relevance. We will explore the underlying principles, the necessary formula, and a step-by-step approach to arrive at the solution. Furthermore, we will discuss the significance of entropy in various chemical and physical processes.

Entropy, in simple terms, is a measure of the number of possible microscopic arrangements or states a system can have. A system with high entropy is more disordered and has more possible arrangements than a system with low entropy. The concept of entropy is central to the second law of thermodynamics, which states that the total entropy of an isolated system can only increase over time or remain constant in ideal cases. This law dictates the direction of spontaneous processes, which tend to proceed towards states of higher entropy. In the context of phase transitions, such as the conversion of liquid water to vapor, the entropy increases significantly because the gaseous state is much more disordered than the liquid state. Understanding entropy changes is crucial in various fields, including chemistry, physics, and engineering, as it helps predict the spontaneity and equilibrium of chemical reactions and physical transformations.

Entropy is often described as a measure of disorder or randomness in a system. A system with high entropy possesses a greater degree of molecular motion and spatial distribution, leading to a more disordered state. Conversely, a system with low entropy exhibits a more ordered and constrained molecular arrangement. The concept of entropy plays a vital role in understanding the spontaneity of physical and chemical processes. According to the second law of thermodynamics, spontaneous processes tend to proceed in the direction that increases the overall entropy of the system and its surroundings. This principle explains why heat flows from hot to cold objects, why gases expand to fill available space, and why chemical reactions proceed to equilibrium. In the context of water vaporization, the transition from the liquid phase to the gaseous phase results in a significant increase in entropy. Liquid water molecules are held together by relatively strong intermolecular forces, restricting their movement. In the gaseous phase, water molecules possess greater kinetic energy and freedom of movement, leading to a more disordered state and, consequently, higher entropy.

Entropy change (ΔS) is a thermodynamic function that measures the change in the degree of disorder or randomness in a system during a process. The entropy change is positive for processes that increase the disorder of the system, such as melting, vaporization, and expansion. Conversely, the entropy change is negative for processes that decrease the disorder of the system, such as freezing, condensation, and compression. The entropy change is a state function, meaning that it depends only on the initial and final states of the system, not on the path taken. The standard entropy change (ΔS°) is the entropy change for a reaction carried out under standard conditions (298 K and 1 atm). Entropy changes are critical in determining the spontaneity of chemical reactions and physical processes. A reaction is more likely to be spontaneous if it leads to an increase in entropy (ΔS > 0) and a decrease in enthalpy (ΔH < 0). The Gibbs free energy change (ΔG) combines both enthalpy and entropy changes to predict spontaneity: ΔG = ΔH - TΔS. A negative ΔG indicates a spontaneous process, while a positive ΔG indicates a non-spontaneous process.

The formula to calculate the entropy change (ΔS) during a phase transition at constant temperature is:

ΔS = Q / T

where:

• ΔS is the entropy change (in J K⁻¹ mol⁻¹)
• Q is the heat absorbed or released during the phase transition (in J mol⁻¹)
• T is the temperature at which the phase transition occurs (in Kelvin)

This equation arises from the definition of entropy change in a reversible process. In a reversible process, the system is always infinitesimally close to equilibrium, and the heat transfer occurs at a constant temperature. The heat absorbed during a phase transition, such as vaporization, is known as the enthalpy of vaporization (ΔHvap). Therefore, the equation can also be written as:

ΔS = ΔHvap / T

This form of the equation is particularly useful when dealing with phase transitions, as the enthalpy of the phase transition is often readily available. In the context of water vaporization, ΔHvap represents the energy required to convert one mole of liquid water into one mole of water vapor at the boiling point. The temperature, T, must be in Kelvin for the equation to be dimensionally consistent. The entropy change, ΔS, is a state function, meaning it depends only on the initial and final states of the system, not on the path taken. This makes the equation applicable regardless of whether the phase transition occurs reversibly or irreversibly. The entropy change calculated using this equation represents the increase in disorder as the substance transitions from a more ordered phase (liquid) to a less ordered phase (gas).

The entropy change formula, ΔS = Q / T, is a cornerstone of thermodynamics, providing a quantitative measure of the change in disorder during a process. This formula is derived from the fundamental definition of entropy in statistical mechanics, which relates entropy to the number of microstates available to a system. In the context of phase transitions, such as the vaporization of water, this formula is particularly useful because it allows us to calculate the entropy change based on the heat absorbed (Q) and the temperature (T) at which the transition occurs. The heat absorbed during a phase transition is often referred to as the latent heat, and in the case of vaporization, it is the enthalpy of vaporization (ΔHvap). The formula highlights that the entropy change is directly proportional to the heat absorbed and inversely proportional to the temperature. This means that a larger amount of heat absorbed leads to a greater increase in entropy, while the same amount of heat absorbed at a higher temperature results in a smaller entropy change. The temperature in the formula must be in Kelvin, as this is the absolute temperature scale, which ensures that the calculations are thermodynamically consistent. The entropy change calculated using this formula is an important thermodynamic parameter that helps in understanding the spontaneity and equilibrium of processes. For example, in the vaporization of water, the positive entropy change contributes to the spontaneity of the process at temperatures above the boiling point.

Given: Heat of vaporization of water = 2260.8 Jg⁻¹ Molar mass of H₂O = 18 g mol⁻¹ Temperature = 100°C

Step 1: Convert the heat of vaporization from Jg⁻¹ to J mol⁻¹

To convert the heat of vaporization from joules per gram (Jg⁻¹) to joules per mole (J mol⁻¹), multiply the given value by the molar mass of water.

Q (J mol⁻¹) = Heat of vaporization (Jg⁻¹) × Molar mass (g mol⁻¹) Q (J mol⁻¹) = 2260.8 Jg⁻¹ × 18 g mol⁻¹ Q = 40694.4 J mol⁻¹

Step 2: Convert the temperature from Celsius to Kelvin

To convert the temperature from Celsius to Kelvin, add 273.15 to the Celsius temperature.

T (K) = T (°C) + 273.15 T (K) = 100°C + 273.15 T = 373.15 K

Step 3: Calculate the entropy change (ΔS)

Now, use the formula ΔS = Q / T to calculate the entropy change.

ΔS = 40694.4 J mol⁻¹ / 373.15 K ΔS = 109.0 JK⁻¹mol⁻¹

Therefore, the entropy change (ΔS) for the conversion of one mole of water to vapor at 100°C is approximately 109.0 JK⁻¹mol⁻¹.

This step-by-step calculation demonstrates the application of the entropy change formula in a practical context. First, it is essential to ensure that the heat of vaporization is expressed in the appropriate units, J mol⁻¹, by multiplying the given value in Jg⁻¹ by the molar mass of the substance. This conversion accounts for the amount of energy required to vaporize one mole of the substance, which is necessary for consistency with the desired units of entropy change (JK⁻¹mol⁻¹). Next, the temperature must be converted from Celsius to Kelvin, as thermodynamic calculations are performed using absolute temperatures. The conversion involves adding 273.15 to the Celsius temperature. Using the formula ΔS = Q / T, the entropy change is then calculated by dividing the heat absorbed during vaporization by the temperature in Kelvin. The result represents the increase in entropy associated with the phase transition. In the case of water vaporization, the positive entropy change indicates an increase in disorder as liquid water transforms into a more disordered gaseous state. The calculated value of 109.0 JK⁻¹mol⁻¹ signifies the magnitude of this increase in disorder per mole of water vaporized at 100°C.

The calculated entropy change (ΔS) of 109.0 JK⁻¹mol⁻¹ for the vaporization of water at 100°C indicates a significant increase in the disorder of the system. This increase in entropy is consistent with the phase transition from a liquid to a gaseous state. In the liquid phase, water molecules are relatively close together and interact through hydrogen bonds, which restrict their movement. When water vaporizes, these intermolecular forces are overcome, and the water molecules gain much greater freedom of movement, leading to a more disordered state. The positive value of ΔS signifies that the vaporization process is accompanied by an increase in the randomness or chaos of the system. This result aligns with the second law of thermodynamics, which states that the total entropy of an isolated system tends to increase over time for spontaneous processes.

The magnitude of the entropy change, 109.0 JK⁻¹mol⁻¹, provides valuable insights into the energy requirements and spontaneity of the vaporization process. A larger entropy change implies a greater increase in disorder, which typically favors the spontaneity of the process. However, the spontaneity of a process also depends on the enthalpy change (ΔH), which represents the heat absorbed or released during the process. In the case of water vaporization, the process is endothermic (ΔH > 0), meaning that energy is required to break the intermolecular forces and convert the liquid to a gas. The Gibbs free energy change (ΔG), which combines both entropy and enthalpy changes (ΔG = ΔH - TΔS), determines the overall spontaneity of the process. At 100°C, the entropy term (TΔS) is large enough to overcome the positive enthalpy change, resulting in a negative ΔG, which indicates that the vaporization of water is spontaneous at this temperature. This analysis underscores the importance of considering both entropy and enthalpy changes when evaluating the spontaneity of a process. The high entropy change associated with water vaporization is a critical factor in many natural phenomena, such as the Earth's water cycle and the regulation of body temperature through sweating.

The entropy change during vaporization has significant implications in various scientific and engineering applications. In chemical engineering, understanding entropy changes is crucial for designing and optimizing processes involving phase transitions, such as distillation and evaporation. For instance, in distillation, the separation of liquids with different boiling points is driven by the difference in their vapor pressures, which is directly related to the entropy change during vaporization. In power generation, the Rankine cycle, which is used in steam power plants, relies on the vaporization and condensation of water to convert thermal energy into mechanical work. The efficiency of the cycle is influenced by the entropy changes during these phase transitions. In environmental science, the entropy change associated with water vaporization plays a key role in the Earth's climate system. The evaporation of water from oceans and other water bodies absorbs a significant amount of heat, which is then released into the atmosphere during condensation. This process helps regulate the Earth's temperature and drives atmospheric circulation patterns. Additionally, the entropy change during vaporization is relevant in biological systems. For example, the evaporation of sweat from the skin is an important mechanism for cooling the body, and the entropy change contributes to the thermodynamic driving force for this process. Therefore, a thorough understanding of entropy changes during phase transitions is essential for addressing a wide range of scientific and technological challenges.

In summary, the entropy change (ΔS) for the conversion of one mole of water to vapor at 100°C is calculated to be 109.0 JK⁻¹mol⁻¹. This calculation involves converting the heat of vaporization to J mol⁻¹, converting the temperature to Kelvin, and applying the formula ΔS = Q / T. The positive value of ΔS indicates a significant increase in disorder during vaporization, which is consistent with the transition from a liquid to a gaseous state. The concept of entropy change is fundamental in thermodynamics and has wide-ranging implications in chemistry, physics, engineering, and environmental science. Understanding entropy changes is crucial for predicting the spontaneity and equilibrium of processes, designing efficient chemical processes, and comprehending natural phenomena. This article has provided a comprehensive guide to calculating entropy change during phase transitions, highlighting its significance and practical applications.

Calculate the entropy change (ΔS) in JK⁻¹mol⁻¹ for the conversion of one mole of ethanol to vapor at its boiling point of 78.3°C, given that the heat of vaporization of ethanol is 841 J/g. [Molar mass of ethanol = 46.07 g/mol]