Welcome to Entropy and Energetics: Understanding Why Reactions Happen
Hello future Chemists! This chapter, Entropy and Energetics, is where we move beyond just heat transfer (\(\Delta H\)) and start asking the ultimate question: "Why does this reaction actually happen?"
You’ve learned that exothermic reactions (\(\Delta H < 0\)) release energy and are often favoured. But what about the endothermic reactions (\(\Delta H > 0\))? Why do some of them occur spontaneously, like ice melting on a warm day? The answer lies in Entropy and Gibbs Free Energy.
Don't worry if this seems abstract at first. We will break down 'disorder' and 'spontaneity' into simple, measurable concepts. Let’s dive in!
Key Takeaway from the Introduction:
Enthalpy (\(\Delta H\)) is only half the story. Entropy (\(\Delta S\)) and Gibbs Free Energy (\(\Delta G\)) determine the true driving force of a chemical change.
Section 1: Understanding Entropy (\(S\)) – The Measure of Disorder
What is Entropy?
Entropy (\(S\)) is a fundamental concept in chemistry that measures the degree of randomness or disorder in a system.
Think of entropy as the universe's tendency toward messiness. A system with high entropy is highly disordered, and a system with low entropy is highly ordered.
Analogy: The Student Bedroom
Imagine your desk:
- Low Entropy: Everything is sorted, books are shelved, and pens are in the pot. (Highly ordered state).
- High Entropy: Books are spread across the floor, papers are piled high, and your mug is nowhere to be found. (Highly disordered state).
Factors Affecting Entropy Change (\(\Delta S\))
A reaction is favoured if the entropy of the system increases (\(\Delta S\) is positive). Several factors lead to an increase in disorder:
- Change of State: Disorder increases as we move from solid to liquid to gas.
Solid (particles locked in place) \(\rightarrow\) Liquid (particles move freely) \(\rightarrow\) Gas (particles move randomly and far apart).
Example: Evaporation (\(H_2O(l) \rightarrow H_2O(g)\)) involves a large positive \(\Delta S\). - Increase in the Number of Particles (Moles): If the products have more moles of gas particles than the reactants, entropy increases significantly.
Example: \(2NH_3(g) \rightarrow N_2(g) + 3H_2(g)\). We go from 2 moles of gas to 4 moles of gas. \(\Delta S\) is positive. - Mixing and Dissolving: Dissolving a solid in a solvent or mixing two gases usually increases disorder.
Example: Dissolving salt (\(NaCl(s) \rightarrow Na^+(aq) + Cl^-(aq)\)). The ordered lattice breaks down, increasing entropy. - Increasing Temperature: As temperature rises, particles move faster, leading to greater thermal disorder and higher entropy.
Quick Review Box: Entropy Signs
If the system becomes more disordered (e.g., more gas produced), \(\Delta S\) is positive (+).
If the system becomes more ordered (e.g., gas turns into liquid), \(\Delta S\) is negative (-).
Section 2: Calculating Standard Entropy Change (\(\Delta S^\ominus\))
Just like with enthalpy, we can calculate the change in entropy for a reaction under standard conditions (denoted by the \(\ominus\) symbol).
The Entropy Calculation Formula
The standard entropy change of a reaction, \(\Delta S^\ominus_{reaction}\), is calculated using the standard molar entropies of the substances involved (\(S^\ominus\)).
\[ \Delta S^\ominus_{reaction} = \sum S^\ominus (products) - \sum S^\ominus (reactants) \]
Key Point on Units!
This is extremely important and a common error point!
Standard Molar Entropy (\(S^\ominus\)) is typically given in Joules per Kelvin per mole:
\[ Units: J K^{-1} mol^{-1} \]
Warning for calculations later: Enthalpy (\(\Delta H\)) is usually measured in \(kJ\). When combining \(\Delta H\) and \(\Delta S\), you MUST convert one of them so they share the same energy unit (usually converting \(\Delta S\) from \(J\) to \(kJ\)).
Step-by-Step Example Calculation
Consider the reaction: \(C(s) + CO_2(g) \rightarrow 2CO(g)\)
Given \(S^\ominus\) values in \(J K^{-1} mol^{-1}\): \(C(s) = 5.7\); \(CO_2(g) = 213.6\); \(CO(g) = 197.6\).
- Calculate Total Entropy of Products:
\(2 \times S^\ominus (CO) = 2 \times 197.6 = 395.2\) - Calculate Total Entropy of Reactants:
\(S^\ominus (C) + S^\ominus (CO_2) = 5.7 + 213.6 = 219.3\) - Calculate \(\Delta S^\ominus\):
\(\Delta S^\ominus = (395.2) - (219.3) = +175.9 J K^{-1} mol^{-1}\)
The positive sign (+175.9) makes sense because we are going from 1 mole of gas (on the left) to 2 moles of gas (on the right), increasing disorder.
Key Takeaway (Entropy Calculation):
\(\Delta S^\ominus\) is calculated by (Products minus Reactants). Always check the units – they are usually in Joules (\(J\)), unlike Enthalpy.
Section 3: Gibbs Free Energy (\(\Delta G\)) – The Ultimate Predictor
In any chemical reaction, two factors drive spontaneity:
- The tendency toward minimum energy (exothermic, \(\Delta H < 0\)).
- The tendency toward maximum disorder (positive entropy change, \(\Delta S > 0\)).
The Gibbs Free Energy Equation
The relationship between Enthalpy, Temperature, and Entropy is defined by the Gibbs Equation:
\[ \Delta G = \Delta H - T\Delta S \]
Where:
- \(\Delta G\): Gibbs Free Energy change (in \(kJ mol^{-1}\)).
- \(\Delta H\): Enthalpy change (in \(kJ mol^{-1}\)).
- \(T\): Absolute Temperature (in Kelvin, K). (Remember: \(K = ^\circ C + 273\)).
- \(\Delta S\): Entropy change (must be in \(kJ K^{-1} mol^{-1}\)).
Crucial Step: Unit Conversion (Do Not Skip!)
Since \(\Delta H\) is usually given in \(kJ\) and \(\Delta S\) in \(J\), you must divide \(\Delta S\) by 1000 before substituting it into the equation.
\[ \text{Converted } \Delta S = \frac{\Delta S \text{ (in } J)}{1000} \]
What Does \(\Delta G\) Tell Us? Spontaneity
Spontaneity means the reaction can happen without continuous external energy input (like constant heating or stirring).
The sign of \(\Delta G\) determines spontaneity:
- If \(\Delta G\) is negative (\(\Delta G < 0\)): The reaction is spontaneous (or feasible) under those conditions. This is the driving force!
- If \(\Delta G\) is positive (\(\Delta G > 0\)): The reaction is non-spontaneous. Energy input is required.
- If \(\Delta G\) is zero (\(\Delta G = 0\)): The system is at equilibrium.
Did You Know?
The term "free energy" refers to the energy actually available in the system to do useful work, like driving an electric current or powering a muscle contraction. The rest is lost to disorder (entropy).
Section 4: The Role of Temperature (T) in Spontaneity
Temperature is critical because it multiplies the entropy term (\(T\Delta S\)). By changing \(T\), we change the relative importance of enthalpy versus entropy.
Let's analyze the four possible combinations of \(\Delta H\) and \(\Delta S\). This analysis is a favourite exam question!
Case 1: Both Favorable
| \(\Delta H\) | Negative (Exothermic) |
| \(\Delta S\) | Positive (Increased Disorder) |
| Result | \(\Delta H\) is negative. \(-T\Delta S\) is negative. |
| Spontaneity | ALWAYS spontaneous at ALL temperatures (\(\Delta G < 0\)). |
Case 2: Both Unfavorable
| \(\Delta H\) | Positive (Endothermic) |
| \(\Delta S\) | Negative (Decreased Disorder) |
| Result | \(\Delta H\) is positive. \(-T\Delta S\) is positive. |
| Spontaneity | NEVER spontaneous at ANY temperature (\(\Delta G > 0\)). |
Case 3: Enthalpy Favored, Entropy Opposed (Low T Spontaneous)
| \(\Delta H\) | Negative (Favorable) |
| \(\Delta S\) | Negative (Unfavorable) |
| Scenario | Spontaneity depends on which term dominates. |
| Spontaneity | Spontaneous only at LOW temperatures, where the favorable \(\Delta H\) term outweighs the unfavorable \(-T\Delta S\) term. |
Case 4: Enthalpy Opposed, Entropy Favored (High T Spontaneous)
| \(\Delta H\) | Positive (Unfavorable) |
| \(\Delta S\) | Positive (Favorable) |
| Scenario | This is why endothermic reactions happen! |
| Spontaneity | Spontaneous only at HIGH temperatures. As \(T\) increases, the favorable \(-T\Delta S\) term (which is negative) eventually becomes larger than the positive \(\Delta H\), making \(\Delta G\) negative. |
Determining the Crossover Temperature (\(T_{eq}\))
For the temperature-dependent cases (3 and 4), there is a temperature point where the reaction switches from spontaneous to non-spontaneous. This is the temperature at which the system is at equilibrium, meaning \(\Delta G = 0\).
To find the crossover temperature (\(T_{eq}\)):
Set \(\Delta G = 0\):
\[ 0 = \Delta H - T_{eq}\Delta S \]
Rearranging gives:
\[ T_{eq} = \frac{\Delta H}{\Delta S} \]
Remember to use consistent units (kJ or J) for \(\Delta H\) and \(\Delta S\) in this calculation! \(T_{eq}\) will be in Kelvin.
Example of Using Crossover Temperature
If you calculate \(T_{eq} = 500 K\):
- If the reaction is Case 4 (\(\Delta H > 0, \Delta S > 0\)): It is spontaneous only when \(T > 500 K\).
- If the reaction is Case 3 (\(\Delta H < 0, \Delta S < 0\)): It is spontaneous only when \(T < 500 K\).
Common Mistake to Avoid
When calculating \(\Delta G\), students often forget to use Kelvin for \(T\) and forget to convert \(\Delta S\) from \(J\) to \(kJ\).
Always ensure \(T\) is in \(K\) and \(\Delta H\) and \(T\Delta S\) are both in \(kJ\) before adding/subtracting them!
Key Takeaway (Spontaneity):
The reaction is spontaneous only if \(\Delta G < 0\). Entropy becomes the dominant factor in driving endothermic reactions at high temperatures.
Chapter Summary and Review
Key Concepts to Master
- Entropy (\(S\)): Measures disorder. Increases when forming gases, mixing, or increasing the number of moles of gas.
- \(\Delta S^\ominus\) Calculation: Products minus Reactants. Units are \(J K^{-1} mol^{-1}\).
- Gibbs Free Energy Equation: \(\Delta G = \Delta H - T\Delta S\).
- Spontaneity: A reaction is spontaneous if \(\Delta G\) is negative.
- Unit Consistency: MUST convert \(\Delta S\) (usually \(J \rightarrow kJ\)) and ensure \(T\) is in Kelvin.
- Crossover Temperature: Calculated using \(T = \Delta H / \Delta S\) (at equilibrium, \(\Delta G = 0\)).
Remember, Chemistry is about balance. This chapter shows that the heat content of a system (\(\Delta H\)) and the disorder of a system (\(\Delta S\)) are constantly competing to determine the fate of a reaction. Keep practising those unit conversions, and you'll master energetics!