6.4.5 - Calculating K from ΔG° or Vice Versa
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Introduction to ΔG° and K
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Today, we're going to discuss the relationship between Gibbs free energy change, ΔG°, and the equilibrium constant, K. Can anyone tell me what is ΔG°?
Is it the energy available to do work under standard conditions?
Exactly! ΔG° represents the spontaneity of a reaction. If ΔG° is negative, the reaction is spontaneous. Now, who can relate this concept to K?
I think K tells us how far a reaction proceeds toward products at equilibrium.
Yes, that's correct! A large K value indicates that products are favored at equilibrium. We represent this relationship with the equation ΔG° = -RT ln K.
What does 'R' and 'T' stand for in that equation?
Good question! R is the ideal gas constant, and T is the temperature in Kelvin. This relationship signifies both spontaneity and extent of reaction.
So, can we use this equation to calculate K from ΔG°?
Absolutely! Let’s summarize this session: ΔG° indicates spontaneity, K shows the extent of the reaction, and they are interconnected through ΔG° = -RT ln K.
Calculating K from ΔG°
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Let’s put this relationship into practice. How would we calculate K if we know ΔG°?
We would rearrange the equation to solve for K!
Exactly! Let’s take an example: if ΔG° = -4700 J mol⁻¹ at 298 K, how would we find K?
We would substitute ΔG° into the equation: -4700 = -RT ln K.
Right! And what would that look like?
Substituting R and T gives us -4700 = -(8.314)(298) ln K?
Perfect! Now how do we isolate ln K?
We divide both sides by -2477.572 to get ln K!
Exactly! Once we find ln K, we can exponentiate to find K. Let’s summarize: to calculate K from ΔG°, we use the equation ΔG° = -RT ln K and rearrange it.
Reverse Calculation: Finding ΔG° from K
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Now let's flip the equation. How can we calculate ΔG° if K is given?
We can start from the same equation, ΔG° = -RT ln K.
That's right! So if K = 6.67, how can we find ΔG° at 298 K?
We’d plug in K into the equation with R and T values.
Yes! So using -RT ln(6.67) should give us the correct ΔG°.
Will we have a negative ΔG° since K is greater than 1?
Exactly! A spontaneous reaction typically yields a negative ΔG°. Let’s summarize: to find ΔG° from K, we rearrange ΔG° = -RT ln K.
Temperature Effect on K and ΔG°
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Let’s talk about how temperature impacts both ΔG° and K. What do you think happens when the temperature changes?
Isn’t K also temperature-dependent?
Absolutely! As temperature changes, so do ΔG° and K. We also have to consider how ΔH° and ΔS° play a role here.
Does that mean we can predict K if we know the enthalpy and entropy changes?
Yes, using the van 't Hoff equation highlights this relationship. Great connections! Can anyone summarize why temperature changes matter?
Temperature changes affect spontaneity and the extent of reactions by altering ΔG° and equilibrium constants.
Excellent summary! Understanding how temperature influences K and ΔG° deepens our grasp of reaction dynamics.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section discusses the fundamental relationship between standard Gibbs free energy change and the equilibrium constant, outlining how to calculate K when ΔG° is known and vice versa. It emphasizes that ΔG° signifies spontaneity while K indicates the extent of a reaction, and also details the dependency of these values on temperature.
Detailed
Detailed Summary
In this section, we delve into the connection between the standard Gibbs free energy change (ΔG°) and the equilibrium constant (K) in chemical reactions. The relationship is articulated through the equation:
ΔG° = -RT ln K
Here, ΔG° represents the change in Gibbs free energy at standard conditions, R is the ideal gas constant, T is the absolute temperature in Kelvin, and K is the equilibrium constant. This relationship provides a quantitative overview of a reaction's spontaneity and its tendency to reach equilibrium:
- If ΔG° < 0 (Negative):
- The reaction is spontaneous, and K is greater than 1, indicating a preference for products.
- If ΔG° > 0 (Positive):
- The reaction is non-spontaneous, and K is less than 1, favoring reactants.
- If ΔG° = 0:
- The system is at equilibrium with K equal to 1, meaning reactants and products are present in comparable concentrations.
The section also touches on the temperature dependency of both ΔG° and K, stipulating that any change in temperature will affect their values. A practical example illustrates how to calculate K from a given ΔG° value at 298 K, further solidifying the concepts presented.
Audio Book
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Fundamental Equation Linking ΔG° and K
Chapter 1 of 3
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Chapter Content
ΔG° = -RT ln K
Where:
● ΔG° is the standard Gibbs free energy change for the reaction (usually in J mol⁻¹ or kJ mol⁻¹). Remember to ensure units consistency between ΔG° and R.
● R is the ideal gas constant (8.314 J K⁻¹ mol⁻¹).
● T is the absolute temperature in Kelvin (K).
● ln K is the natural logarithm of the equilibrium constant (K). K can be Kc or Kp, depending on the reaction, but the equation uses a dimensionless K (as equilibrium constants are truly dimensionless when activities are used).
Detailed Explanation
This equation serves as a bridge between Gibbs free energy (ΔG°), which tells us whether a reaction is spontaneous, and the equilibrium constant (K), which describes the extent of the reaction at equilibrium.
- Understanding the symbols: Each variable has specific meanings. ΔG° indicates the change in Gibbs free energy, R is the universal gas constant which quantifies energy, T is the temperature in Kelvin, and ln K is the logarithmic form of the equilibrium constant K.
- Interpreting the equation: If we know ΔG°, we can find K by rearranging the equation to K = e^(-ΔG°/RT). This demonstrates how spontaneous reactions (with ΔG° < 0) correspond to a K value greater than 1, indicating that products are favored at equilibrium.
Examples & Analogies
Imagine a hill where ΔG° represents the effort to climb it. If ΔG° is negative, it’s like having a slide down – it's easy to reach the bottom (products) quickly. If K is greater than 1, it means there are more final products than reactants, similar to ending up with more marbles at the bottom of a funnel than at the top.
Calculating K from ΔG° Example
Chapter 2 of 3
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Chapter Content
Example: Calculating K from ΔG°
For the reaction 2NO₂(g) ⇌ N₂O₄(g), ΔG° = -4.7 kJ mol⁻¹ at 298 K. Calculate the value of Kp.
1. Convert ΔG° to Joules:
ΔG° = -4.7 kJ mol⁻¹ = -4700 J mol⁻¹
2. Use the relationship ΔG° = -RT ln Kp:
-4700 J mol⁻¹ = -(8.314 J K⁻¹ mol⁻¹)(298 K) ln Kp
3. Solve for ln Kp:
-4700 = -2477.572 ln Kp
ln Kp = -4700 / -2477.572 ≈ 1.897
4. Solve for Kp:
Kp = e^(1.897) ≈ 6.67
The value of Kp being greater than 1 is consistent with a negative ΔG°, indicating that the dimerization of NO₂ is spontaneous under standard conditions at 298 K and favours product formation at equilibrium.
Detailed Explanation
In this example, we demonstrate how to apply the equation to find the equilibrium constant K from the standard Gibbs free energy change ΔG°:
- Unit Conversion: First, we convert ΔG° from kilojoules to joules because R is in J K⁻¹ mol⁻¹, ensuring our units align.
- Applying the Key Equation: Next, we substitute the known values (ΔG°, R, and T) into the equation ΔG° = -RT ln Kp.
- Solving for Kp: We rearrange the equation to find ln Kp, then exponentiate to get Kp. The resulting Kp indicates that the reaction strongly favors product formation, confirming that the reaction is spontaneous.
Examples & Analogies
Think of K as a measure of popularity. If a new product (like a smartphone) is well-received (ΔG° is negative), it quickly gains a large market share (Kp > 1). By using the equation, we can predict its market success based on initial expectations (ΔG°). Just like predicting market trends from consumer reactions, we can predict chemical behaviors from Gibbs free energy.
Calculating ΔG° from K Example
Chapter 3 of 3
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Chapter Content
Example: Calculating ΔG° from K
Suppose we know K = 2.0 at 298 K for the reaction A ⇌ B.
1. Use the relationship ΔG° = -RT ln K:
ΔG° = -(8.314 J K⁻¹ mol⁻¹)(298 K) ln(2.0)
2. Calculate ln(2.0):
ln(2.0) ≈ 0.693
3. Substitute and solve:
ΔG° = -2477.572 J mol⁻¹ (0.693) ≈ -1715 J mol⁻¹
Converting to kJ, ΔG° = -1.715 kJ mol⁻¹. This negative value indicates the reaction is spontaneous under these conditions.
Detailed Explanation
This example illustrates how to derive ΔG° from a known K value:
- Using the Relationship: We start with the equation ΔG° = -RT ln K, where R and T are constants.
- Calculating ln K: We find the natural logarithm of the equilibrium constant K to use in our equation.
- Final Calculation: After substitution, we calculate ΔG°, which helps determine if the reaction is spontaneous. A negative ΔG° confirms it is likely to occur naturally.
Examples & Analogies
Imagine trying to assess the risk of rain based on a weather forecast (K). If K indicates a high probability of rain (greater than 1), you would conclude it’s more likely to rain (negative ΔG°), just as we derive a reaction's spontaneity from the equilibrium constant.
Key Concepts
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ΔG° represents the energy change signaling whether a reaction is spontaneous.
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K indicates how far a reaction proceeds toward products at equilibrium.
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The relationship ΔG° = -RT ln K links energy and equilibrium.
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Temperature affects both ΔG° and K, altering spontaneity and reaction extent.
Examples & Applications
Calculating K for the reaction 2NO₂(g) ⇌ N₂O₄(g) using ΔG° = -4.7 kJ mol⁻¹ at 298 K.
Finding ΔG° for a reaction with K = 6.67 using the equation ΔG° = -RT ln K.
Memory Aids
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Rhymes
If ΔG° is low, the reaction will glow, K will go high—products in tow.
Stories
A chemist named Delta was curious about spontaneous reactions. One day, while measuring Gibbs free energy, they discovered that whenever ΔG° was negative, K was more than one, and products flourished, creating a success story!
Memory Tools
Remember: 'GKE' - Gibbs Leads to K's Extent—ΔG° helps define K in experiments.
Acronyms
K.E.Y. - K (Equilibrium) = ΔG° + R.T (Temperature) factors.
Flash Cards
Glossary
- Gibbs Free Energy Change (ΔG°)
The change in Gibbs free energy for a reaction under standard conditions, indicating spontaneity.
- Equilibrium Constant (K)
A numerical value representing the ratio of concentrations (or partial pressures) of products to reactants at equilibrium.
- Ideal Gas Constant (R)
A constant used in equations involving gases, typically 8.314 J K⁻¹ mol⁻¹.
- Natural Logarithm (ln)
The logarithm to the base e, often used in thermodynamic equations.
- Spontaneity
The tendency of a reaction to proceed without external influence.
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