Question

Determine the free energy change, \(\Delta \mathrm{G}\), for transforming liquid water at \(100^{\circ} \mathrm{C}\) and 1 atm. to vapor at the same conditions. \(\Delta \mathrm{H}=9720\) cal (molar enthalpy of vaporization).

Short answer

The free energy change, \(\Delta G\), for transforming liquid water at \(100^{\circ} \mathrm{C}\) and 1 atm to vapor at the same conditions is \(\Delta G = 9720 \,\text{cal}\).

Step-by-step solution

Convert given temperature to Kelvin

The given temperature is in Celsius, we need to convert it to Kelvin for our calculations. The conversion formula is: Temperature in Kelvin (K) = Temperature in Celsius (°C) + 273.15 So, temperature in Kelvin is: \(T = 100 + 273.15 = 373.15 K\)

Use the enthalpy change and Clausius-Clapeyron equation to find the entropy change

The Clausius-Clapeyron equation is given as: \[\ln\left(\frac{P_{2}}{P_{1}}\right) = \frac{-\Delta H}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)\] Since the pressure is the same for both states (1 atm), we have \(P_1 = P_2\), which means the left side of the equation becomes 0. Therefore, we have: \(0 = \frac{-\Delta H}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)\) Since the equation equals to 0, we can discard \(\Delta H\) and \(R\). Thus, we are left with the following: \(0 = \frac{1}{T_2} - \frac{1}{T_1}\) By the statement "at the same conditions," we also have \(T_1 = T_2\). Therefore, the entropy change, \(\Delta S\), for this process is 0.

Calculate the free energy change

Now that we have found the entropy change, we can use the relationship between free energy change, enthalpy change, and entropy change to find the free energy change \(\Delta \mathrm{G}\). This relationship is given as: \(\Delta \mathrm{G} = \Delta \mathrm{H} - T\Delta \mathrm{S}\) Since we found that \(\Delta \mathrm{S} = 0\), the formula simplifies to: \(\Delta \mathrm{G} = \Delta \mathrm{H}\) Using the given enthalpy change, we can find the free energy change: \(\Delta \mathrm{G} = 9720 \,\text{cal}\)

Present the final result

The free energy change for transforming liquid water at \(100^{\circ} \mathrm{C}\) and 1 atm to vapor at the same conditions is: \(\Delta \mathrm{G} = 9720 \,\text{cal}\)

Key concepts

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is a fundamental principle in thermodynamics that describes the relationship between the pressure and temperature of two phases of a substance in equilibrium, such as the liquid and vapor states of water. With this equation, students can calculate how the vapor pressure of a substance changes with temperature—a crucial aspect for understanding phase transitions.

This equation is particularly useful for predicting how much pressure is required to boil or condense a substance at a specific temperature. In simple terms, it tells us how temperature and vapor pressure are related for a pure substance during phase changes. The equation is expressed as:
\[\ln\left(\frac{P_{2}}{P_{1}}\right) = \frac{-\Delta H}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)\]
where:

• \(P_1\) and \(P_2\) are the pressures at temperatures \(T_1\) and \(T_2\), respectively
• \(\Delta H\) represents the enthalpy change of the phase transition
• \(R\) is the universal gas constant

Understanding this equation aids in comprehension of how energy is absorbed or released during a phase change.

Enthalpy of Vaporization

The enthalpy of vaporization, symbolized as \(\Delta H_{vap}\), is the amount of energy required to convert a given quantity of a substance from a liquid into a vapor without changing its temperature. This thermodynamic quantity is a measure of the energy barrier that molecules need to overcome to escape from the liquid phase and enter the gas phase.

The enthalpy of vaporization is a crucial concept in calculations involving phase changes, especially when substances transition from liquid to gas, such as the boiling of water. This energy is absorbed from the surroundings, and thus it is considered endothermic. The enthalpy of vaporization can be experimentally determined and is often provided in calories per mole or joules per mole. For instance, the exercise provided an enthalpy of vaporization of \(9720 \text{cal}\) for water to vaporize.

Gibbs Free Energy

Gibbs free energy, or \(\Delta G\), is an essential concept in thermodynamics that represents the maximum amount of work that can be performed by a thermodynamic process at constant temperature and pressure. It is a state function that helps predict the direction of chemical reactions and phase transitions.

It is calculated using the formula:\[\Delta G = \Delta H - T\Delta S\]
where:

• \(\Delta H\) is the change in enthalpy
• \(T\) is the temperature in Kelvin
• \(\Delta S\) is the change in entropy

In a process where the temperature and pressure are constant, if \(\Delta G\) is negative, the process occurs spontaneously. On the other hand, if \(\Delta G\) is positive, the process is nonspontaneous and requires input of energy to occur. Understanding the relationship between Gibbs free energy, enthalpy, and entropy allows us to predict the feasibility of a reaction or process under given conditions.

Thermodynamics

Thermodynamics is the branch of physical science that deals with the relations between heat and other forms of energy. It describes how energy is transferred within systems and how it affects matter, allowing us to understand phenomena such as heat engines, refrigerators, and reactions in living organisms.

One of the key themes in thermodynamics is the law of conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form into another. There are four main laws of thermodynamics that define physical quantities, such as temperature, energy, and entropy, and illustrate how they are interrelated and how they can predict the behavior of a given system.
In summary, thermodynamics helps us understand the energy changes that accompany physical and chemical processes, providing the theoretical underpinning for many practical applications in engineering, chemistry, biology, and environmental science.