Entropy

Spontaneous vs. Non-Spontaneous Processes

• Thermodynamics deals with spontaneous and non-spontaneous changes.
• Spontaneous processes occur naturally (e.g., a skier going downhill).
• Non-spontaneous processes require external intervention (e.g., a skier going uphill without assistance).
• Examples of spontaneous processes:
  • Diffusion of a gas.
  • Flow of heat from hot to cold.
  • Marbles thrown in the air spreading apart.
  • Heating a liquid until it boils.
• Reverse processes are non-spontaneous (marbles reforming into a package, heat flowing from cold to hot).
• Dynamite pieces will never reform into the original stick after explosion.

Entropy (S)

• Spontaneous processes are accompanied by an increase in the disorder of the universe, which is entropy.
• Entropy is represented by the capital letter $S$, indicating it's a state function.
• Entropy is a measure of disorder.
• Example: Dynamite stick is more ordered than its fragmented pieces after explosion.
• Living organisms are highly ordered, but nature tends to increase disorder (entropy) in the universe.

Processes with Decreasing Entropy

• Condensation: Gas to liquid conversion decreases entropy (decrease in disorder).
• Other examples: crystallization, precipitation, freezing.
• Relating entropy directly to spontaneity is complex.

Entropy and States of Matter

• Entropy of a solid < entropy of a liquid << entropy of a gas.
• Gases have the highest disorder and kinetic energy.
• Solids and liquids are condensed phases.

Units and Absolute Entropy Values

• Entropy is a thermodynamic quantity measured in joules per Kelvin mole ( $J/K \cdot mol$ ).
• Enthalpic contributions to energy changes are generally larger than entropic contributions (by about three orders of magnitude).
• Unlike enthalpy ($\Delta H$) values, entropy values are absolute and positive, according to the third law of thermodynamics.
• The standard state for gases is one atmosphere, but for thermodynamics, the standard state is $25$ degrees Celsius.

Third Law of Thermodynamics

• Classical physics: At 0 Kelvin, a perfect crystalline solid with no molecular motion has zero entropy.
• All other entropy values are relative to this benchmark.
• Quantum mechanics suggests that complete cessation of molecular motion is impossible due to the Heisenberg uncertainty principle.

Statistical Nature of Entropy

• Statistical mechanics: Highest entropy corresponds to the system with the most microstates (individual statistical states).
• Example: Gas molecules in a container with a valve separating an empty side. Opening the valve leads to diffusion and an increase in entropy.
• Although uniform gas distribution may seem more ordered, each gas molecule has more available locations (microstates), making it more disordered.
• Entropy drives processes like gas laws, heat flow, and diffusion.
• Molecules on the hot side have higher kinetic energy; heat flow to the cold side increases microstates for every molecule.

Entropy Diagram

• Processes increasing entropy:
  • Solid melting to liquid.
  • Liquid boiling/evaporating to vapor.
  • Dissolving a compound in a solvent to form a solution.
  • Heating a system (T1 to T2) increases kinetic energy and microstates (even in solids, promoting molecular vibration/rotation).

Entropy Change in Chemical Reactions

• $\Delta S<em>{reaction} = S</em>{products} - S_{reactants}$
• \Delta S > 0 if the moles of gas increase.
• \Delta S < 0 if the moles of gas decrease.
• $\Delta S$ is negligible if there's no net change in gas molecules.
• Gases being evolved or consumed significantly affect $\Delta S$.

Precipitation Reaction Exception

• Precipitation reactions often show \Delta S < 0 (products more ordered than reactants).
• Example: Silver ions reacting with chloride ions to form silver chloride precipitate.
• Despite decreasing entropy, these reactions can be spontaneous.

Example Problems: Predicting the Sign of $\Delta S$

• General rule: Compare gas moles on reactant vs. product sides, except for precipitation.
• A superscripted zero means $\Delta S$ is at standard conditions.

Example 1

• $N<em>2(g) + 2H</em>2(g) \rightarrow NH_3(g)$
• 3 moles of gas become 1 mole: \Delta S < 0

Example 2

• $2CO(g) + O<em>2(g) \rightarrow 2CO</em>2(g)$
• 2 moles of gas going to 2 moles of gas: $\Delta S \approx 0$

Example 3

• $2N<em>2O</em>5(g) \rightarrow 4NO<em>2(g) + O</em>2(g)$
• 5 moles of gas become 6 moles: \Delta S > 0

Example 4

• $C(s) + O<em>2(g) \rightarrow CO</em>2(g)$
• 1 mole of gas yields 2 moles: \Delta S > 0

Example 5

• $Ag^+(aq) + Cl^-(aq) \rightarrow AgCl(s)$
• Precipitation reaction: \Delta S < 0

More Examples

• Two-chamber experiment: gas molecules distributed evenly has higher entropy. \Delta S < 0. Unfavorable process
• Salt dissolving: reverse precipitation, \Delta S > 0, favorable process
• Gas reactants to liquid products: \Delta S < 0, not favorable

Calculating Delta S for Chemical Reactions

• If absolute entropy values are known:
• $\Delta S<em>{reaction} = \sum S</em>{products} - \sum S_{reactants}$
• $\Delta S$ Formula for a Reaction:
  • For $aA + bB \rightarrow cC + dD$
  • $\Delta S = [c \cdot S(C) + d \cdot S(D)] - [a \cdot S(A) + b \cdot S(B)]$
  • Standard conditions: one atmosphere, $25$ degrees Celsius.

Example: Calculate Delta S

• Reaction: $CH<em>4(g) + 2O</em>2(g) \rightarrow CO<em>2(g) + 2H</em>2O(l)$
• Using Appendix III, look up entropy values for each compound (all positive due to the third law).
• $\Delta S = [1 \cdot S(CO<em>2) + 2 \cdot S(H</em>2O)] - [1 \cdot S(CH<em>4) + 2 \cdot S(O</em>2)]$
• $\Delta S = [1 \cdot (213.6) + 2 \cdot (69.9)] - [1 \cdot (186.3) + 2 \cdot (205.0)] = -242.8 \frac{J}{K}$

Spontaneity and Entropy Contradiction

• Statements: Entropy must increase for spontaneous processes, yet many spontaneous processes have decreasing entropy.
• Examples: Freezing, condensation, crystallization.
• The second law of thermodynamics resolves this. In the next video.