6.3.2 - Scenario 2: Calculating Equilibrium Concentrations/Partial Pressures from Initial Conditions and K (ICE Tables)
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Introduction to ICE Tables
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Today, we are going to learn how to calculate equilibrium concentrations using the concept of ICE Tables, which stands for Initial, Change, and Equilibrium. Can anyone tell me what they think this means?
Is it like a way to track how much of each substance we have before and after the reaction?
Exactly! The 'Initial' row captures what we start with, 'Change' shows how much of each substance is consumed or produced, and 'Equilibrium' gives us the final amounts. Now, letβs look at a specific example, the decomposition of PClβ .
Setting Up the ICE Table
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For the reaction PClβ (g) β PClβ(g) + Clβ(g), suppose we start with 1.00 mol of PClβ in a 10.0 dmΒ³ container. Can we first find the initial concentration?
I think we just divide the number of moles by the volume. So, 1.00 mol divided by 10.0 dmΒ³ gives us 0.100 mol/dmΒ³, right?
Correct! Now, how do we fill in the rest of the ICE table based on that information?
We'll put 0 in the PClβ and Clβ columns for Initial because there are none yet.
Exactly! Letβs now write down the changes we would expect as the reaction reaches equilibrium.
Calculating Changes in Concentration
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Letβs introduce 'x' as the change in concentration for PClβ . What would the changes look like in our table?
PClβ would decrease by x, and PClβ and Clβ would each increase by x!
That's correct! How do we express this in our ICE table?
We fill it in: 0.100 - x for PClβ , and x for both PClβ and Clβ.
Well done! Finally, we can write the equilibrium constant expression for this reaction.
Using the Kc Expression
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Now we need to write the Kc expression based on our ICE table. Can someone do that for me?
Kc = [PClβ][Clβ] divided by [PClβ ]!
Excellent! And if Kc is 0.024, what do we do next?
We substitute the equilibrium expressions into Kc and solve for x!
Perfect! As you calculate x, remember we can often work with quadratics here if it gets tricky.
Final Calculations and Results
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After solving the quadratic, how do we calculate the equilibrium concentrations?
We just plug x back into our equilibrium expressions!
Exactly! If we use the approximation that x is small when K is low, we save time and simplify calculations. Always check if x is less than 5% of the initial concentration to be sure.
Got it! This is really helpful for figuring out equilibrium concentrations!
Great! Remember, using the ICE table structure organizes our approach and leads us to accurate results.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, we explore the ICE (Initial, Change, Equilibrium) method for calculating equilibrium concentrations or partial pressures from initial conditions and the equilibrium constant. This method systematically lays out the initial amounts, changes that occur as the system reaches equilibrium, and the final equilibrium concentrations based on K values.
Detailed
Scenario 2: Calculating Equilibrium Concentrations Using ICE Tables
In the study of chemical equilibrium, being able to predict the concentrations or partial pressures of substances at equilibrium is crucial. When given initial concentrations (or amounts) and the equilibrium constant (K), we can find the equilibrium concentrations using an ICE tableβa structured approach for organizing the information.
ICE Table Structure
- Initial (I): The concentrations of all species before any reaction occurs.
- Change (C): The amount of change in concentration of reactants and products as the system approaches equilibrium, typically represented by a variable (x).
- Equilibrium (E): The final concentrations at equilibrium calculated from the changes.
Example: Decomposition of PClβ
To illustrate the ICE table method, let's consider the decomposition of phosphorus pentachloride (PClβ ):
PClβ (g) β PClβ(g) + Clβ(g)
Suppose we have the equilibrium constant Kc = 0.024 at a certain temperature, and we start with 1.00 mol of PClβ in a 10.0 dmΒ³ container.
- Compute the initial concentrations:
- initial [PClβ ] = 1.00 mol / 10 dmΒ³ = 0.100 mol/dmΒ³
- Products start at 0 mol/dmΒ³.
- Set up the ICE table:
| Species | Initial (I) | Change (C) | Equilibrium (E) |
|---|---|---|---|
| PClβ (g) | 0.100 | -x | 0.100 - x |
| PClβ(g) | 0.0 | +x | x |
| Clβ(g) | 0.0 | +x | x |
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Write the Kc expression:
Kc = [PClβ][Clβ] / [PClβ ] - Substitute expressions into Kc and solve for x (the change in concentration):
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With Kc = 0.024,
0.024 = (x)(x)/(0.100 - x). - Simplifying leads to a quadratic equation. Once solved, we can find the equilibrium concentrations by substituting x back into the equilibrium expressions.
This structured method not only simplifies calculations but provides clarity on how changes in concentrations affect equilibrium, allowing chemists to make informed predictions about reaction outcomes.
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Introduction to ICE Tables
Chapter 1 of 6
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Chapter Content
When you are given initial concentrations and the value of K, you need to determine how the concentrations will change to reach equilibrium. This is commonly done using an ICE (Initial, Change, Equilibrium) table.
Detailed Explanation
ICE Tables are useful tools for calculating the changes in concentrations of reactants and products as a system approaches equilibrium. The table is structured into three rows: Initial (I), Change (C), and Equilibrium (E). In the Initial row, we note the starting concentrations of all chemicals involved. The Change row indicates how those concentrations will change as the reaction progresses towards equilibrium, often represented with a variable like 'x'. Finally, the Equilibrium row calculates the concentrations at equilibrium based on the initial amounts and the changes.
Examples & Analogies
Think of an ICE table like a recipe for a cake. The initial ingredients represent the reactants you start with. As the mixing and baking process progresses, some ingredients change into cake (the products) until you have a delicious treat ready to serve (the equilibrium state).
Example Calculation: PClβ Decomposition
Chapter 2 of 6
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Chapter Content
Consider the decomposition of PClβ : PClβ (g) β PClβ(g) + Clβ(g). At a certain temperature, Kc = 0.024. If 1.00 mol of PClβ is placed in a 10.0 dmΒ³ container, calculate the equilibrium concentrations.
Detailed Explanation
In this example, we start with 1.00 mol of PClβ in a 10 dmΒ³ container, giving us an initial concentration of 0.100 mol dmβ»Β³ for PClβ . The initial concentrations of PClβ and Clβ are both zero because they haven't formed yet. We set up our ICE table where we denote the change in the concentration of PClβ as '-x'. When PClβ reacts, it decreases in concentration while PClβ and Clβ increase by 'x'. This leads to a new equilibrium concentration for each substance. We then write the equilibrium expression based on the reaction and substitute our equilibrium concentrations into the expression to set it equal to K and solve for 'x.' Finally, we calculate the equilibrium concentrations for all substances.
Examples & Analogies
Imagine a balloon filled with only air (PClβ ). When you start to let the air out, the balloon's volume decreases and the air inside it (which represents the gas in our reaction) finds a new balance. When you find balance againβwith some air now outside the balloon (representing PClβ and Clβ)βthat illustrates reaching equilibrium.
Setting Up the ICE Table and Calculating Equilibrium
Chapter 3 of 6
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Chapter Content
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Initial Concentrations:
[PClβ ] = 1.00 mol / 10.0 dmΒ³ = 0.100 mol dmβ»Β³
[PClβ] = 0 mol dmβ»Β³
[Clβ] = 0 mol dmβ»Β³ -
Set up the ICE Table: Let 'x' be the change in concentration of PClβ
.
Concentration (mol dmβ»Β³) PClβ PClβ Clβ
Initial (I) 0.100 0 0
Change (C) -x +x +x
Equilibrium (E) 0.100-x x x
Detailed Explanation
Initially, we calculate the concentration of PClβ as 0.100 mol dmβ»Β³ since we have 1.00 mol in a 10 dmΒ³ container. For PClβ and Clβ, their initial concentrations are zero as they haven't formed yet. We use a variable 'x' to represent how much of PClβ dissociates into PClβ and Clβ. In the ICE table, for the Initial row, we place our starting concentrations. For the Change row, we mark a decrease of 'x' for PClβ and an increase of 'x' for PClβ and Clβ. Consequently, the Equilibrium row reflects the concentrations after the reaction reaches equilibrium.
Examples & Analogies
This process is similar to deciding how many slices of a cake (PClβ in our example) to eat at a party. Initially, you have an entire cake (0.100 mol of PClβ ). As you and your friends eat slices (the change), the remaining cake decreases while the consumed pieces can be equated to the number of happy faces (PClβ and Clβ) around you, leading to an eventual calm party atmosphere (equilibrium).
Equilibrium Constant Expression and Solving for 'x'
Chapter 4 of 6
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Chapter Content
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Write the Kc expression:
Kc = [PClβ][Clβ] / [PClβ ] -
Substitute equilibrium expressions into Kc and solve for x:
0.024 = (x)(x) / (0.100 - x)
0.024 (0.100 - x) = xΒ²
0.0024 - 0.024x = xΒ²
xΒ² + 0.024x - 0.0024 = 0
This is a quadratic equation (axΒ² + bx + c = 0). Use the quadratic formula:
x = [-b Β± sqrt(bΒ² - 4ac)] / 2a
Detailed Explanation
The equilibrium constant expression relates the concentrations of products to those of reactants at equilibrium. For the decomposition of PClβ , we express Kc in terms of PClβ and Clβ concentrations as they are the products, and PClβ is the reactant. After substituting the values from our ICE table into the Kc expression, we simplify and rearrange the equation, which ultimately leads us to a quadratic equation in the form axΒ² + bx + c = 0. We can solve this equation using the quadratic formula to find the value of 'x' which tells us how much of PClβ has converted into PClβ and Clβ.
Examples & Analogies
Imagine you're setting up a balance scale (the equilibrium constant) with weights (the concentrations). As you adjust the weights of cakes (PClβ and Clβ), you want to find out how much cake has been transferred from one side (PClβ ) to balance it effectively. The quadratic equation acts as the detailed plan to find the right amount of cake needed to achieve perfect equilibrium on the scale.
Finding the Positive Solution for 'x' and Equilibrium Concentrations
Chapter 5 of 6
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x = [-0.024 Β± sqrt((0.024)Β² - 4(1)(-0.0024))] / 2
x = [-0.024 Β± sqrt(0.000576 + 0.0096)] / 2
x = [-0.024 Β± sqrt(0.010176)] / 2
Two possible values for x:
xβ = (-0.024 + 0.10087) / 2 = 0.0384 (approx.)
xβ = (-0.024 - 0.10087) / 2 = -0.0624 (approx.)
Since concentration cannot be negative, we choose x = 0.0384 mol dmβ»Β³.
- Calculate Equilibrium Concentrations:
[PClβ ] = 0.100 - 0.0384 = 0.0616 mol dmβ»Β³
[PClβ] = 0.0384 mol dmβ»Β³
[Clβ] = 0.0384 mol dmβ»Β³
Detailed Explanation
When we solve the quadratic equation, we end up with two values for 'x': one positive and one negative. Since concentration cannot be negative, we discard the negative solution. We keep the positive solution, x = 0.0384 mol dmβ»Β³, which represents the amount of reactant converted to products. Next, we calculate the equilibrium concentrations by subtracting 'x' from the initial concentration of PClβ and keeping 'x' for the concentrations of PClβ and Clβ. Thus, we find the equilibrium concentrations for all species involved in the reaction.
Examples & Analogies
Returning to our cake analogy, when we're adjusting the recipe (solving for 'x'), we discard any impossible solutions (like negative cake amounts). The positive amount tells us the final slices shared at the party, giving us the final cake pieces present for everyone to enjoy, leading to that delightful moment when everyone relishes the cake (the equilibrium state).
Approximation Method for Small K values
Chapter 6 of 6
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Chapter Content
Approximation Method: If the value of K is very small (typically K < 10β»Β³ or 10β»β΄) and the initial concentration of reactants is relatively large, you can often make the approximation that 'x' is negligible compared to the initial concentration. For example, in 0.100 - x, if x is very small, 0.100 - x β 0.100. This simplifies the calculation by avoiding the quadratic formula. After solving for x with the approximation, you must check if x is less than 5% of the initial concentration. If it is, the approximation is valid. If not, the quadratic formula must be used. In the example above, x (0.0384) is not negligible compared to 0.100 (it's 38.4%), so the approximation would not be valid.
Detailed Explanation
When K is small compared to the initial concentration of reactants, we can simplify the calculation process. This is because the amount of reactant that converts to products (x) will be very small relative to the initial amount. Therefore, we can assume that the initial concentration changes only very slightly, allowing us to approximate calculations without using complex equations. However, itβs important to check if this assumption holds true: if 'x' is less than 5% of the initial concentration, the approximation can be used safely. If 'x' turns out to be significant, we need to revert to using the quadratic formula to ensure accuracy.
Examples & Analogies
Think about making a tiny adjustment to a large car engine (the reaction). If you're only making a small tweak to the throttle (the reaction shifting slightly), it's often easier to keep everything else the same, especially if it works well (ignoring that tiny adjustment). But if your tweak becomes significant, leading to a noticeable change in performance, you need to look closely at the entire engine (use the quadratic equation) to avoid any issues. This is how we approach calculations in chemistry conservatively.
Key Concepts
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ICE Table: A structured format used to organize equilibrium calculations.
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Equilibrium Concentration: Concentrations of reactants and products at equilibrium.
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Kc Expression: Mathematical representation of the equilibrium constant based on concentrations.
Examples & Applications
For a reaction A β B, if initial concentrations are given and K is known, use an ICE table to find equilibrium concentrations.
For the decomposition reaction PClβ β PClβ + Clβ with Kc = 0.024, set up the ICE table to predict the equilibrium concentrations from given initial conditions.
Memory Aids
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Rhymes
In the ICE table we embark, we track the change, not just a lark. Initial, change, and equilibrium too, these three steps help us see what's true.
Stories
Imagine a party where reactants are dancing around. At first, theyβre shy and alone, but as they mingle, PClβ splits off to invite PClβ and Clβ to the fun, creating equilibriumβa balanced dance!
Memory Tools
Remember ICE as 'Ice Caps' to think of Initial Concentration first, Change as what drips, and Equilibrium as the steady state left when the drips stop.
Acronyms
Use I.C.E. (Initial - Change - Equilibrium) to remember the order you find equilibrium concentrations.
Flash Cards
Glossary
- ICE Table
A visual tool used to organize the initial concentrations, the changes that occur, and the equilibrium concentrations in a chemical reaction.
- Equilibrium Constant (K)
A value that represents the ratio of concentrations of products to reactants at equilibrium, indicating the extent of a reaction.
- Initial Concentration
The concentration of a substance before any reaction occurs.
- Change (x)
The variable used to denote the change in concentration of reactants and products as they reach equilibrium.
- Equilibrium Concentration
The concentration of a substance when the reaction has reached a state of balance and does not change over time.
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