Question

The gas-phase reaction of \(\mathrm{H}_{2}\) with \(\mathrm{CO}_{2}\) to produce \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{CO}\) has \(\Delta \mathrm{H}=11 \mathrm{kJ}\) and \(\Delta \mathrm{S}=41 / \mathrm{K} .\) Is the reaction spontaneous at 298.15 \(\mathrm{K} ?\) What is \(\Delta G ?\)

Short answer

ΔG = -1.22415 kJ. The reaction is spontaneous at 298.15 K.

Step-by-step solution

Identify the Given Values

Given values are ΔH = 11 kJ (Enthalpy Change) and ΔS = 41 J/K (Entropy Change). Note that ΔS needs to be converted into kJ/K to match the units of ΔH.

Convert Entropy Change to kJ/K

ΔS = 41 J/K * 0.001 = 0.041 kJ/K.

Write the Equation for Gibbs Free Energy

The equation for Gibbs Free Energy is \[ΔG = ΔH - TΔS\] where T is the temperature in Kelvin (298.15 K in this case).

Substitute Values into the Equation

ΔG = 11 kJ - (298.15 K * 0.041 kJ/K).

Calculate the Product of T and ΔS

Compute 298.15 K * 0.041 kJ/K = 12.22415 kJ.

Compute ΔG

Subtract the value obtained in the previous step from ΔH: ΔG = 11 kJ - 12.22415 kJ = -1.22415 kJ.

Determine Spontaneity

If ΔG is negative, the reaction is spontaneous. In this case, ΔG = -1.22415 kJ, which is negative, so the reaction is spontaneous at 298.15 K.

Key concepts

enthalpy change

Enthalpy change, denoted as \(\triangle \text{H}\), is a measure of the total heat content in a system. This value indicates whether a reaction absorbs heat (endothermic) or releases heat (exothermic). When \(\triangle \text{H}\) is positive, the reaction is endothermic, meaning it absorbs heat from its surroundings. Conversely, when \(\triangle \text{H}\) is negative, the reaction is exothermic, releasing heat.
In the given problem, \(\triangle \text{H} = 11 \text{kJ}\). This positive value tells us that the reaction absorbs heat, making it endothermic. Understanding the enthalpy change helps us know the energy needed or released during the reaction process. If we were to apply this to everyday life, think about how an instant ice pack works. It absorbs heat from its surroundings, making your injury feel cooler – that's an endothermic process.

entropy change

Entropy change, represented as \(\triangle \text{S}\), measures the disorder or randomness in a system. A positive entropy change means the system's disorder is increasing, while a negative entropy change means it's becoming less disordered.
In the exercise, we're given \(\triangle \text{S} = 41 \text{J/K}\), which we convert to \(0.041 \text{kJ/K}\). This positive value implies that the reaction leads to an increase in disorder. For example, melting ice increases disorder as the structured ice crystals turn into freely moving water molecules.
Understanding entropy change helps us predict the distribution of energy and matter in a chemical process. It reveals insights into how spontaneous a reaction may be when considering energy dispersal.

spontaneity of reactions

The spontaneity of reactions is determined by Gibbs Free Energy, \(\triangle \text{G}\). The equation used is \[ \triangle \text{G} = \triangle \text{H} - T \triangle \text{S} \] If \(\triangle \text{G}\) is negative, the reaction is spontaneous, meaning it can occur without any external input of energy. If \(\triangle \text{G}\) is positive, the reaction is non-spontaneous and requires energy to proceed.
In our case, we calculated \(\triangle \text{G} = -1.22415 \text{kJ}\), indicating that the reaction is spontaneous at 298.15 K.
This concept is significant because it helps predict whether a reaction will occur under certain conditions, aiding in the design and understanding of chemical processes, such as industrial manufacturing and biological systems. \(\triangle \text{G}\) essentially combines the effects of enthalpy and entropy to give a complete picture of a reaction's favorability.

thermodynamics

Thermodynamics is the branch of physical science that deals with the relations between heat and other forms of energy, like work. It involves key principles and laws that describe how energy is transferred and how it affects matter. Understanding thermodynamics is crucial in chemistry and engineering because it governs how and why reactions occur.
The three main laws of thermodynamics are:

• The First Law (Law of Energy Conservation): Energy cannot be created or destroyed, only transformed from one form to another.
• The Second Law: The entropy of an isolated system always increases over time.
• The Third Law: As the temperature of a system approaches absolute zero, the entropy of the system approaches a constant minimum.

In the given exercise, thermodynamics principles help us calculate and understand how the enthalpy, entropy, and temperature interact to determine Gibbs Free Energy, thereby predicting the spontaneity of the reaction.
By mastering these concepts, students can better analyze and conclude the energy changes involved in chemical reactions and real-world processes, like why ice melts or why engines produce work.