Question

Calculate the free energy change for this reaction at \(25^{\circ} \mathrm{C} .\) Is the reaction spontaneous? $$\mathrm{C}_{3} \mathrm{H}_{8}(g)+5 \mathrm{O}_{2}(g) \longrightarrow 3 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)$$ $$\Delta H_{\mathrm{mm}}^{\circ}=-2217 \mathrm{k} \mathrm{k} ; \quad \Delta S_{\mathrm{mn}}^{\circ}=101.1 \mathrm{J} / \mathrm{K}$$

Short answer

\(\Delta G^{\circ} = -2247147.3 \mathrm{J}\), the reaction is spontaneous at \(25^\circ C\).

Step-by-step solution

Understanding Gibbs Free Energy

Gibbs free energy (\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text(\text{\text{\text(\text{\text{\text{\text{\text{\text{\text{\text{\text{\text(\text{\text{\text{\text{\text(\Delta G}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}^{\circ}$) can be calculated using the formula: \(\Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ}\), where \(\Delta H^{\circ}\) is the change in enthalpy, \(T\) is the temperature in Kelvin, and \(\Delta S^{\circ}\) is the change in entropy.

Convert Temperature to Kelvin

Convert the given temperature from Celsius to Kelvin by adding 273.15: \[ T (K) = 25^\circ C + 273.15 = 298.15 K \]

Calculate Gibbs Free Energy Change

Use the formula for Gibbs free energy along with the previously determined temperature in Kelvin and the given \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) to calculate \(\Delta G^{\circ}\): \[ \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \] Substitute the given values: \[ \Delta G^{\circ} = (-2217 \mathrm{k} \mathrm{J}) - (298.15 \mathrm{K}) \times (101.1 \mathrm{J}/\mathrm{K}) \] Convert kilojoules to joules for \(\Delta H^{\circ}\) by multiplying by 1000: \[ \Delta G^{\circ} = (-2217000 \mathrm{J}) - (298.15 \mathrm{K}) \times (101.1 \mathrm{J}/\mathrm{K}) \] Now perform the calculations.

Perform the Calculations

Multiply the temperature in Kelvin by the change in entropy, then subtract that from the change in enthalpy: \[ \Delta G^{\circ} = -2217000 \mathrm{J} - 298.15 \mathrm{K} \times 101.1 \mathrm{J}/\mathrm{K} \] \[ \Delta G^{\circ} = -2217000 \mathrm{J} - (298.15 \times 101.1) \mathrm{J} \] \[ \Delta G^{\circ} = -2217000 \mathrm{J} - 30147.3 \mathrm{J} \] \[ \Delta G^{\circ} = -2247147.3 \mathrm{J} \]

Determine Spontaneity

A negative value for \(\Delta G^{\circ}\) indicates that the reaction is spontaneous under standard conditions.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy. In essence, it studies the movement of energy and how energy transforms from one form to another. The fundamental laws of thermodynamics (zeroth, first, second, and third laws) lay the groundwork for understanding how energy cycles within a system, be it an engine, the Earth's atmosphere, or a chemical reaction.

• The Zeroth Law establishes that if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
• The First Law, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another.
• The Second Law asserts that the total entropy of a closed system can never decrease over time; it can only stay constant or increase, providing the underpinning for understanding spontaneous processes.
• The Third Law states that as the temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant minimum.

The applications of thermodynamics are broad and include systems ranging from the microscopic, like quantum gases, to astrophysical objects like stars.

Spontaneous Reaction

A spontaneous reaction is a process that occurs without the need for continual external influence or energy to maintain it; it can proceed on its own given the initial conditions. It's important to note that a spontaneous reaction isn't necessarily quick; it might be so slow as to be imperceptible, like the rusting of iron, or it can be rapid, like the explosive reaction of gasoline with oxygen. In thermodynamics, spontaneity often depends on changes in enthalpy, entropy, and temperature of the system.

To determine if a reaction is spontaneous, we look at the Gibbs free energy change (\(\Delta G\)), following this criterion: if \(\Delta G\) is negative, the process is spontaneous. If \(\Delta G\) is positive, the process is non-spontaneous under the conditions. If \(\Delta G\) is zero, the system is at equilibrium.

Enthalpy Change

Enthalpy change (\(\Delta H\)) is the measure of heat absorbed or released by a system at constant pressure. It is a central concept in chemical thermodynamics used to predict whether reactions will release heat (exothermic) or absorb heat (endothermic).

• An exothermic reaction has a negative \(\Delta H\) because it releases heat to its surroundings.
• An endothermic reaction has a positive \(\Delta H\) because it absorbs heat from its surroundings.

To calculate the enthalpy change of a reaction, we often refer to standard enthalpy changes, which are measured under standard conditions (298 K and 1 atm pressure), and denoted as \(\Delta H^\circ\). In the case of the combustion of propane in the original exercise, the negative enthalpy change indicates that heat is released, and hence, the reaction is exothermic.

Entropy Change

Entropy (\(S\)) is a measure of disorder or randomness in a system. The change in entropy (\(\Delta S\)) during a reaction indicates the degree to which the disorder of the system changes. In thermodynamics, the second law implies that for a spontaneous process, the entropy of the universe (the system plus its surroundings) must increase.

A positive \(\Delta S\) means that the disorder of the system increases through the reaction process, while a negative \(\Delta S\) implies a decrease in disorder. Entropy change contributes to the determination of spontaneity in a reaction; a higher entropy often favors spontaneous reactions, especially in the presence of a high enough temperature. In the given exercise, the positive entropy change suggests an increase in disorder as the reaction progresses from reactants to products.