Question

In this problem, you will calculate the differences in the chemical potentials of ice and supercooled water and of steam and superheated water, all at 1 atm pressure shown schematically in Figure \(8.1 .\) For this problem, \(S_{H_{2} O, s}^{\circ}=48.0 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}, S_{H_{2}}^{\circ} 0, l=70.0 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1},\) and \(S_{H_{2}, O, g}^{\circ}=188.8 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}\). a. By what amount does the chemical potential of water exceed that of ice at \(-2.25^{\circ} \mathrm{C} ?\) b. By what amount does the chemical potential of water exceed that of steam at \(102.25^{\circ} \mathrm{C} ?\)

Short answer

At -2.25°C, the chemical potential of water exceeds that of ice by approximately 5960 J/mol. At 102.25°C, the chemical potential of water exceeds that of steam by approximately 44528 J/mol.

Step-by-step solution

Convert temperature to Kelvin

To use the Gibbs-Helmholtz equation, we need to convert the given temperature from Celsius to Kelvin. To do this, add 273.15 to the given temperature: \(T = -2.25 + 273.15 = 270.90 K\)

Find the difference in entropy

Calculate the difference in entropy between supercooled water and ice using the given values for \(S_{H_2 O, s}^\circ\) and \(S_{H_2 O, l}^\circ\): \(\Delta S = S_{H_2 O, l}^\circ - S_{H_2 O, s}^\circ = 70.0 - 48.0 = 22.0 \, \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}\)

Calculate the difference in chemical potential

Use the Gibbs-Helmholtz equation to find the difference in chemical potentials: \(\Delta G = \Delta H - T \Delta S\) Since both phases are under the same pressure (1 atm), we can assume that the change in enthalpy, \(\Delta H\), is negligible. Thus, \(\Delta G \approx - T \Delta S\). Plugging in the values we have: \(\Delta G \approx - (270.90 \, \mathrm{K})(22.0 \, \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}) = -5960 \, \mathrm{J} \mathrm{mol}^{-1}\) So the chemical potential of water exceeds that of ice at -2.25°C by approximately 5960 J/mol. b. Calculate the difference in chemical potential of water and steam at 102.25°C:

Convert temperature to Kelvin

Convert the given temperature from Celsius to Kelvin: \(T = 102.25 + 273.15 = 375.40 \, \mathrm{K}\)

Find the difference in entropy

Calculate the difference in entropy between superheated water and steam using the given values for \(S_{H_2 O, l}^\circ\) and \(S_{H_2 O, g}^\circ\): \(\Delta S = S_{H_2 O, g}^\circ - S_{H_2 O, l}^\circ = 188.8 - 70.0 = 118.8 \, \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}\)

Calculate the difference in chemical potential

Use the Gibbs-Helmholtz equation to find the difference in chemical potentials: \(\Delta G = \Delta H - T \Delta S\) Again, since both phases are under the same pressure (1 atm), we can assume that the change in enthalpy, \(\Delta H\), is negligible. Thus, \(\Delta G \approx - T \Delta S\). Plugging in the values we have: \(\Delta G \approx - (375.40 \, \mathrm{K})(118.8 \, \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}) = -44528 \, \mathrm{J} \mathrm{mol}^{-1}\) So the chemical potential of water exceeds that of steam at 102.25°C by approximately 44528 J/mol.

Key concepts

Understanding the Gibbs-Helmholtz Equation

In the realm of thermodynamics, the Gibbs-Helmholtz equation establishes a vital relationship between the enthalpy (Hcirc) of a system, its temperature (T), and its free energy (Gcirc). Essentially, this equation helps us understand how the energy within a system changes with temperature when the system is at constant pressure. The formula is expressed as:
\[circ = Hcirc- TScirc\]
In practical terms, the Gibbs-Helmholtz equation enables us to calculate the change in chemical potential (circ) of substances, thereby indicating their stability and spontaneity of reactions or transformations. As seen in the exercise, the equation simplifies when the change in enthalpy (H) is negligible, which is common during phase transitions at constant pressure.

Practical Application of the Equation

By applying the Gibbs-Helmholtz equation to the textbook problem, we discover how phase changes, such as water to ice or water to steam, have quantifiable energy changes associated with them. This allows us to analyze the feasibility of these processes under various conditions. Furthermore, when entropy (S) values are provided, the equation can uncover the change in free energy without requiring knowledge of the system's enthalpy.

• Calculation of temperature in Kelvin for accuracy in formula usage
• Determination of the entropy difference between two phases
• The application of the equation to estimate free energy changes and chemical potentials

Thus, this equation not only serves as a conceptual bridge connecting thermodynamic functions but also as a practical tool for solving real-life chemical puzzles.

Entropy - The Measure of Disorder

Entropy, symbolized by (S), is a fundamental concept in thermodynamics, described as a measure of the randomness or disorder within a system. The second law of thermodynamics dictates that for any spontaneous process, the entropy of the universe increases. It is a state function, meaning its value depends only on the current state of the system, not on how it got there.

Entropy's Role in Phase Transitions

When we examine phase transitions, entropy plays a key role. During the transition from solid to liquid, or liquid to gas, the system's entropy increases due to the higher degree of freedom and disorder of the particles. Conversely, when a gas condenses to a liquid or a liquid freezes to a solid, the entropy of the system decreases owing to the reduced motion and increased orderliness of the particles.
In the exercise solution, the difference in entropy between two phases of water (solid and liquid, or liquid and gas) directly impacts the chemical potential. When temperatures are given, the entropy changes can be used to determine how much the chemical potential of one phase exceeds another. Notably, entropy is also a critical factor when considering the environmental conditions (like temperature and pressure) under which a substance will naturally change phase.

• Increased entropy during melting and evaporation
• Decreased entropy during freezing and condensation
• Impact of entropy on natural phase changes of substances

Understanding entropy is crucial for students to grasp why substances behave the way they do during phase changes and how this concept is integral to solving thermodynamic problems.

Phase Transitions and Chemical Potential

A phase transition is the transformation of a substance from one state of matter to another, such as solid to liquid (melting), liquid to gas (boiling), or the reverse processes (freezing and condensation). These transitions occur at specific temperatures and pressures, known as the melting point and boiling point respectively.

Chemical Potential During Transitions

The chemical potential (circ) of a substance can be thought of as the energy change associated with the addition or removal of a small amount of the substance from a system at constant temperature and pressure. It reflects the 'willingness' of the substance to undergo phase change.
In the given exercise, by understanding how the chemical potential of water compares to that of ice or steam, one can infer the direction in which the phase transition naturally occurs. For instance, if the chemical potential of water is higher than that of ice, then the natural propensity would be for the water to freeze, releasing energy and creating a more stable state at the given condition.

• Melting, boiling, freezing, and condensation as phase transitions
• Temperatures and pressures at which transitions occur
• Role of chemical potential in determining the spontaneity of phase changes

Studying phase transitions provides insight into the intrinsic properties of materials and is pivotal in fields ranging from meteorology to culinary arts. It explains the conditions necessary for a substance to stay in a certain phase or to change to a different one, which is directly related to differences in chemical potential.