Question

At \(373 \mathrm{~K}\), steam and water are in equilibrium and \(\Delta H=40.98 \mathrm{~kJ} \mathrm{~mol}^{-1}\). What will be \(\Delta S\) for conversion of water into steam? \(\mathrm{H}_{2} \mathrm{O}_{(l)} \rightarrow \mathrm{H}_{2} \mathrm{O}_{(g)}\) (a) \(109.8 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) (b) \(31^{-1} \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) (c) \(21.93 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) (d) \(326 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\)

Short answer

\(\Delta S\) for the conversion of water into steam at 373 K is approximately \(109.89 \, \text{J K}^{-1} \text{mol}^{-1}\).

Step-by-step solution

Understanding the Gibbs Free Energy Equation

To find the entropy change (\(\Delta S\)) for the conversion of water into steam, use the Gibbs Free Energy equation under equilibrium conditions: \(\Delta G = \Delta H - T\Delta S\). When steam and water are in equilibrium, \(\Delta G = 0\).

Substituting the Known Values

Knowing that \(\Delta G = 0\) at equilibrium, substitute the known values of \(\Delta H\) and \(T\) into the Gibbs equation to solve for \(\Delta S\): \(0 = 40.98 \, \text{kJ mol}^{-1} - (373 \, \text{K}) \cdot \Delta S\). Convert \(\Delta H\) to \(\text{J mol}^{-1}\) by multiplying by 1000 to get \(40980 \, \text{J mol}^{-1}\).

Solving for \(\Delta S\)

Rearrange the equation to solve for \(\Delta S\): \(\Delta S = \frac{\Delta H}{T}\). Substituting \(\Delta H\) as \(40980 \, \text{J mol}^{-1}\) and \(T\) as 373 K, we get \(\Delta S = \frac{40980}{373} \approx 109.89 \, \text{J K}^{-1} \text{mol}^{-1}\).

Key concepts

Understanding Gibbs Free Energy

Gibbs Free Energy, represented by the symbol \( G \), is a thermodynamic quantity that measures the maximum amount of reversible work that can be performed by a system at a constant temperature and pressure. It is a central concept in the field of thermodynamics and provides vital information about the spontaneity of processes and chemical reactions. In essence, if the change in Gibbs Free Energy (\( \Delta G \)) for a process is negative, the process can occur spontaneously, while a positive \( \Delta G \) indicates non-spontaneity.

The equation \( \Delta G = \Delta H - T\Delta S \) elegantly intertwines the enthalpy change (\( \Delta H \)), the temperature in Kelvin (\( T \)), and the entropy change (\( \Delta S \)), highlighting how changes in energy and disorder affect a process. Thermodynamics often deals with complex systems, but grasping the Gibbs equation's applications, one can decipher the energetic feasibility of reactions, such as the phase transition from water to steam.

Entropy Change for Phase Transition

Entropy is a measure of disorder or randomness in a system, and its symbol is \( S \). Entropy change (\( \Delta S \)) is particularly significant during phase transitions where a substance goes from one state of matter to another, such as solid to liquid or liquid to gas. This change implicates a substantial shift in the order of the system.

When water turns into steam at its boiling point, the molecules shift from a relatively ordered liquid state to a much more disordered gaseous state, thus increasing the system's entropy. The amount of entropy change is crucial for understanding the thermodynamic properties of the phase transition and can be calculated using the equilibrium condition \( \Delta G = 0 \) in the Gibbs Free Energy equation. Calculating \( \Delta S \) during the phase transition of water into steam involves quantifying the amount of energy per temperature unit that gets distributed among the system's particles in this process.

Phase Transition of Water

The phase transition of water from liquid to vapor (boiling) is a common physical transformation with significant implications in thermodynamics. At the boiling point of water, 373 K (100°C), the liquid and gas phases coexist in equilibrium. The latent heat of vaporization (\( \Delta H \)) is the amount of heat required to convert one mole of water to steam without changing the temperature.

Understanding how the transfer of heat relates to the disorder in the system during this phase transition is vital for various applications ranging from meteorology to engineering. The precise calculation of entropy change during this transition using the Gibbs Free Energy equation informs us about the energy dynamics involved and is fundamental in the field of physical chemistry.

Applying Thermodynamics to Calculate Entropy Change

In thermodynamics, calculations such as finding the entropy change for converting water into steam are not just theoretical exercises; they have practical significance. For instance, understanding the entropy change is essential in designing and improving steam engines and power plants.

By working through the given problem step by step, we arrive at a calculated entropy change of approximately 109.89 J K^-1 mol^-1, which reflects the increased disorder as water molecules escape into the gaseous state. Not only does this figure help in the understanding of the specific process of boiling water, but it also serves as a fundamental concept in broader thermodynamic cycles where heat engines and refrigerators operate based on the principles of energy transfer and entropy changes.