Question

Calculate the entropy change that occurs when 1.00 mol of steam is converted to liquid water at \(100^{\circ} \mathrm{C}\) in a reversible process. \(\left(q_{\mathrm{vap}}=40.7 \mathrm{kJ} /\right.\)mol)

Short answer

The entropy change is approximately -109.1 J/mol·K.

Step-by-step solution

Understand the Given Data

We are given that 1.00 mol of steam is converted to liquid water at 100°C in a reversible process. The heat of vaporization ( q_{ ext{vap}} ) is 40.7 kJ/mol. Our goal is to find the change in entropy for this process.

Define Entropy Change Formula

Entropy change ( riangle S) for a process can be calculated using the formula: \[ riangle S = \frac{q_{ ext{rev}}}{T} \] where q_{ ext{rev}} is the heat exchanged reversibly and T is the temperature in Kelvin.

Convert Temperature to Kelvin

Since the process occurs at 100°C, we need to convert this temperature to Kelvin by adding 273.15: \[ T = 100 + 273.15 = 373.15 \text{ K} \]

Calculate Entropy Change

Apply the formula for entropy change: \[ riangle S = \frac{q_{ ext{rev}}}{T} = \frac{-40.7 \, \text{kJ/mol}}{373.15 \, \text{K}} \]Note that the heat is negative because it's released when steam condenses to water.Convert 40.7 kJ to J (since we're dealing with J/ ext{K}): 40.7 kJ = 40700 J.Then calculate: \[ riangle S = \frac{-40700 \, \text{J/mol}}{373.15 \, \text{K}} \approx -109.1 \, \text{J/mol} \cdot \text{K} \]

Conclusion

The change in entropy for the conversion of 1.00 mol of steam to liquid water at 100°C is approximately -109.1 J/mol·K. This negative value indicates a decrease in entropy, which is typical for a condensation process.

Key concepts

Reversible Process

A reversible process is an idealized type of process in thermodynamics where the system changes state in such a way that the process can be reversed by an infinitesimal change in a variable. This means no energy is lost to the surroundings as the process happens.
- In reality, no real-world process is truly reversible because there is always some energy lost to the environment as waste heat.

• In a reversible process, the system is always in thermodynamic equilibrium with its surroundings.
• Everything happens gradually, ensuring all the internal and external pressures and temperatures remain balanced.

This concept is important because reversible processes demonstrate optimal energy efficiency, providing a benchmark for the efficiency of real-world systems. In our exercise, the conversion of steam to liquid water occurs as a reversible process, emphasizing minimal entropy production.

Heat of Vaporization

The heat of vaporization, often denoted as \(q_\mathrm{vap}\), is the amount of heat required to convert a substance from a liquid to a gas at its boiling point without changing its temperature.
- For water, at 100°C, the heat of vaporization is 40.7 kJ/mol.

• This is the energy needed to break the intermolecular forces in the liquid phase, allowing molecules to escape into the gaseous state.
• When steam condenses back into liquid water, the same amount of energy is released.

In our exercise, we are dealing with the condensation of steam, which is the reverse of vaporization. Therefore, when steam converts to liquid water, 40.7 kJ/mol of energy is released, impacting the entropy of the system.

Temperature Conversion Kelvin

Temperature conversion to Kelvin is essential when dealing with thermodynamic processes, as Kelvin is the standard unit of temperature in science.
- To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.

• This is because 0 K represents absolute zero, the point at which no thermal energy remains in a substance.
• In the context of the exercise, 100°C is converted to 373.15 K.

Using Kelvin in thermodynamic equations ensures accuracy as it avoids negative temperatures that could complicate calculations, especially in entropy changes where temperature appears in the denominator.

Condensation Process

The condensation process is a phase change where a gas transforms into a liquid. This transformation involves the release of heat, making it an exothermic process.
- In our problem, the condensation of steam to liquid water releases energy.

• Condensation results in a decrease in entropy because the gas phase generally has higher entropy than the liquid due to greater disorder.
• The entropy change was calculated as approximately -109.1 J/mol·K, reflecting this decrease.

Understanding condensation is crucial in numerous natural and industrial processes, such as in the formation of clouds or the operation of heat exchangers in power plants.