One of the most important aspects of equilibrium is that it is the point at which a reaction “settles” and concentrations will not change because the rate of the forward reaction and reverse reaction will equal each other. However, what happens if we know the equilibrium constant for a reaction, but don’t know what the equilibrium concentrations for a reaction are? Well, we have to do some math!
In this section, you’ll learn how to use an ICE Box (sometimes called a RICE Box) to solve for equilibrium concentrations and learn some of the math behind one of AP Chemistry’s most important mathematical techniques.
ICE Boxes Explained
When solving for an equilibrium concentration, we use an ICE Box. The “I” stands for “initial”, the “C” stands for “change”, and the “E” stands for “equilibrium”. If you ever see RICE Box, the "R" stands for "reaction." Both techniques are the same!
These boxes are used to show the initial concentrations, change in concentration, and final equilibrium concentrations for a reaction.
Here’s what an ICE Box looks like with the reaction CH₃COOH ⇌ CH₃COO⁻ + H⁺ (K = 1.8 * 10⁻⁵):
| Reaction | CH₃COOH | CH₃COO⁻ | H⁺ |
|---|---|---|---|
| Initial | [CH₃COOH]₀ | 0 | 0 |
| Change | -x | +x | +x |
| Equilibrium | [CH₃COOH]₀ - x | x | x |
| Where: |
- [CH₃COOH]₀ represents the initial concentration of acetic acid
- x represents the amount of CH₃COOH that dissociates at equilibrium
- K = 1.8 × 10⁻⁵ = (x × x)/([CH₃COOH]₀ - x) = x²/([CH₃COOH]₀ - x)
Filling in an ICE Table
Let’s start filling this in! Let’s suppose we are starting with an initial CH₃COOH concentration of 1 M (initial concentrations are typically given!). Because we’re starting at the very beginning before any reaction has occurred, the concentration of the products, [CH₃COO⁻] and [H⁺], are 0:
| Reaction | CH₃COOH | CH₃COO⁻ | H⁺ |
|---|---|---|---|
| Initial | 1 M | 0 | 0 |
| Change | -x | +x | +x |
| Equilibrium | 1 - x | x | x |
| Next, let’s think about how these quantities change. When this reaction occurs, some of our reactants will start converting into products. However, because our K is low, not all of the reactant is going to turn into products, so we don’t know how much we form. Therefore in the change row, we use x as a variable to denote this change. For reactants, we lose nx (where n is the stoichiometric coefficient), and for each product, we gain nx. At equilibrium, we add together the initial and change to get our final concentrations! Let’s see this in the table: |
- When Q<K, the reaction proceeds with net consumption of reactants and generation of products
- When Q>K, the reaction proceeds with net consumption of products and generation of reactants
- When Q=K, the system is at dynamic equilibrium
| Reaction | CH₃COOH | CH₃COO⁻ | H⁺ |
|---|---|---|---|
| Initial | 1 M | 0 | 0 |
| Change | -x | +x | +x |
| Equilibrium | 1 - x | x | x |
| Note that K is low and we are displaying change, we subtract x for the reactants to note a loss of concentration and we add x for the products to note an increase in concentration. We're basically showing what happens during a reaction using this table and describing the initial and final conditions. |
You may be wondering what happens after filling in the ICE table. Now that we have solved for the equilibrium concentrations, we can use our equilibrium constant, K, to solve for x.
Solving for K with Equilibrium Concentrations
Finally, we can plug these equilibrium concentrations into our formula for the equilibrium constant to solve:
K = [CH₃COO⁻][H⁺] / [CH₃COOH]
K was given as 1.8 * 10⁻⁵ and we can plug the values in the "Equilibrium" row of the ICE table in:
K = 1.8 * 10⁻⁵ = [x][x] / [1-x] = x² / 1-x
From here, we could multiply both sides by (1-x) and use the quadratic formula, but remember that our K value is super super small. When K is small, we aren’t going to have produced much product at equilibrium meaning that 1 - x is incredibly close to 1. We’ll elaborate a bit on this later, but for this reason, we can approximate 1 - x to be 1 to make our calculations easier.
Note that 1-x is a tiiiny bit less than 1 in reality, but because x << 1 (x is much less than 1), we can make the approximation. It basically makes such a little difference in our calculations that we can eliminate it. Therefore, we can write the following equation to solve for x:
x² / 1 = 1.8 * 10⁻⁵
x = √(1.8 * 10⁻⁵) = 0.0042.
Yay! We solved for x. Now we know that the equilibrium concentrations of [CH₃COO⁻] and of [H+] are 0.0042 (because they’re x according to our ICE table). Similarly, we can say that the equilibrium concentration of [CH₃COOH] is 1 - 0.0042, which is 0.9958. This is pretty close to 1, which makes sense because we’re not going to lose too much reactant to form products at equilibrium. Note that the units in ICE boxes will typically be Molarity to represent concentration.
When to Use ICE Tables
This strategy can be applied to any reaction you’re given as long as you have:
- The equilibrium constant
- Some initial concentration(s) (you don’t always start at the beginning, but any ungiven concentrations will be 0) You can also use ICE tables to solve for initial concentrations, and in turn, Q. There are a ton of calculations that get unlocked once you learn this technique, but we'll get into that later in this unit and in unit eight. Unit eight is all about acids and bases, and ICE tables are particularly useful when working with reactions that involve a weak acid or weak base.
5% Approximation: Eliminating X
Many students struggle with the idea of simply dropping x from the denominator when using an ICE box. Let’s think about this idea completely isolated from ICE Boxes or even equilibrium. Let’s suppose we have a number that is super tiny, like, astronomically tiny. Let’s set a number x = 0.0000001.
If we want to find a value like 3+x, we could manually add 3 and 0.0000001 to get 3.0000001, but note that this value is basically 3. Sure, it’s slightly off but when we’re doing calculations like x²/(3+x) = 1.43 * 10⁻³ (these are just made-up numbers), being able to approximate 3+x to 3 makes our calculations way easier. Typically, scientists say that we can make this approximation when it does not lead to an error larger than 5%, which is why it's called the 5% approximation or 5% rule.
However, on the AP Chemistry exam, you can almost always make this approximation and it makes your equilibrium concentration calculations worlds easier because you won’t have to deal with a quadratic. The one approximation you want to be careful not to make is approximations that directly affect x. For example, even though it’s technically valid as an approximation, saying “4x = 0” would screw up calculations way more because they would always equal 0 which ruins the point of finding equilibrium concentrations!
Basically, in any ICE Box problem (as far as AP Chemistry is concerned), we can approximate a + x or a - x to be just a because x is much much less than a.
Practicing with ICE Boxes
Consider the following reaction:
H₂CO₃ ⇌ HCO₃⁻ + H⁺ (K = 4.3 x 10⁻⁷)
At equilibrium, what is the concentration of [HCO₃⁻] if the reaction began with an initial H₂CO₃ concentration of 1.2 M?
Let’s write out an ICE Box to solve this problem!
| Reaction | H₂CO₃ | HCO₃⁻ | H+ |
|---|---|---|---|
| Initial | 1.2 M | 0 M | 0 M |
| Change | -x | +x | +x |
| Equilibrium | 1.2 - x | x | x |
| Note that the stoichiometric coefficients are one in this reaction as well, so there is no number in front of the x in the change row of the ICE table. Say the stoichiometric coefficient of H₂CO₃ was 2, how would that affect the above ICE table? Well, the "change" for H₂CO₃ would be -2x instead of just -x. Now, let's get back to the math for this problem and solve for the concentration of HCO₃⁻. |
4.3 * 10⁻⁷ = [x][x] / [1.2 - x] ≈ x² / 1.2 (this step is using the 5% rule)
x² = 1.2 * (4.3 * 10⁻⁷) = 5.16 * 10⁻⁷
[HCO₃⁻] = x = √(5.16 * 10⁻⁷) = 0.0007.
Vocabulary
The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.
| Term | Definition |
|---|---|
| balanced reaction | A chemical equation in which the number of atoms of each element is equal on both sides of the equation. |
| chemical species | A distinct chemical entity such as an atom, molecule, or ion that participates in a chemical reaction. |
| concentration | The amount of solute dissolved in a given volume of solution, typically expressed in molarity or other units of amount per volume. |
| dynamic equilibrium | A state of equilibrium in which forward and reverse reactions continue to occur at equal rates, maintaining constant macroscopic properties. |
| equilibrium | The state in which the forward and reverse reaction rates are equal, resulting in constant concentrations or partial pressures of reactants and products. |
| equilibrium constant | A numerical value that expresses the ratio of products to reactants at equilibrium, indicating the extent to which a reaction proceeds. |
| forward reaction | The reaction pathway in which reactants are converted to products. |
| initial conditions | The starting concentrations or partial pressures of reactants and products before a reaction reaches equilibrium. |
| net consumption | The overall decrease in the amount of a substance as a result of a chemical reaction. |
| partial pressures | The individual pressure exerted by each gas in a mixture of gases at equilibrium. |
| reaction quotient | A value calculated using the same expression as the equilibrium constant but using current (non-equilibrium) concentrations or partial pressures. |
| reverse reaction | The reaction that proceeds from products back to reactants, opposite to the direction written in the balanced chemical equation. |
Frequently Asked Questions
What is the equilibrium constant and how do I use it to find concentrations?
The equilibrium constant K (Kc for concentrations, Kp for pressures) is a number that relates product and reactant amounts at equilibrium: K = [products]^(coeff)/[reactants]^(coeff). To find equilibrium concentrations: 1) Write the balanced equation and the K expression. 2) Make an ICE table (Initial, Change, Equilibrium) using given initial concentrations or partial pressures. 3) Express equilibrium terms with a variable (x) for the change, substitute into the K expression, and solve for x (you may get a quadratic—use the quadratic formula or the small-x approximation if K is very small/large). 4) Calculate equilibrium concentrations and check Q (reaction quotient) to confirm direction: Q
How do you calculate equilibrium concentrations when you only know the initial amounts?
Start with the balanced equation, write initial concentrations (or partial pressures), then build an ICE table: Initial, Change (use +/− stoichiometric x), Equilibrium (initial ± x). Write the equilibrium expression for Kc (or Kp if given pressures) and substitute the equilibrium concentrations in terms of x. Solve the resulting equation for x (often a quadratic). If K is very large or very small you can use the quadratic approximation (drop the “± x” in the denominator if x is <5% of the initial value) and then check that the approximation is valid. Always compute Q with initial values to predict direction (Q
I'm confused about Q vs K - what's the actual difference and when do I use each one?
K is the equilibrium constant for a given reaction at a specific temperature—it tells you the ratio [products]^coeff/[reactants]^coeff when the system is at equilibrium (use Kc for concentrations, Kp for partial pressures). Q (the reaction quotient) has the same algebraic form as K but uses the current (not necessarily equilibrium) concentrations or pressures. How to use them: - Calculate Q from initial conditions. - If Q < K → net reaction goes forward (reactants consumed, products formed). - If Q > K → net reaction goes reverse (products consumed). - If Q = K → system is at dynamic equilibrium. In problems use Q to predict direction of shift, then set up an ICE table and use K to solve for equilibrium concentrations (or partial pressures). For AP-style questions, practice ICE + quadratic approximation and know when to use Kc vs Kp. For guided practice see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and try extra problems at (https://library.fiveable.me/practice/ap-chemistry).
Why does the reaction go forward when Q is less than K?
Q is just the current value of the equilibrium expression; K is the value at equilibrium. If Q < K, the ratio of products to reactants is too small compared with the equilibrium ratio, so the net forward reaction is favored—the system will convert reactants into products until Q rises to equal K. In kinetic terms the forward rate > reverse rate, so product concentrations increase and reactant concentrations decrease. Thermodynamically, ΔG = ΔG° + RT ln Q: when Q < K, ln Q < ln K and ΔG < 0, so the forward reaction is spontaneous (drives toward equilibrium). In problems you’ll see this used with ICE tables: compute Q from initial concentrations, compare to K, then set up changes (use x) and solve for equilibrium concentrations (7.7.A in the CED). For more examples and practice, see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and unit resources (https://library.fiveable.me/ap-chemistry/unit-7). If you want, I can walk through a quick ICE-table example.
What does it mean when Q equals K and how do I know if my system is at equilibrium?
Q = K means the reaction is at dynamic equilibrium: the forward and reverse rates are equal, so concentrations (or partial pressures) don’t change over time even though both reactions still occur. To test whether your system is at equilibrium, calculate the reaction quotient Q from the current concentrations/pressures and compare it to K (use Kc for concentrations, Kp for partial pressures): - If Q < K → net reaction goes forward (reactants → products) until equilibrium is reached. - If Q > K → net reaction goes reverse (products → reactants). - If Q = K (within experimental/significant-figure error) → system is at equilibrium. In practice use an ICE table to get Q from initial values, then compare to the given K to predict the direction or confirm equilibrium (CED LO 7.7.A). For more worked examples and steps for ICE and quadratic approximations, see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and more practice problems at (https://library.fiveable.me/practice/ap-chemistry).
How do I set up an ICE table for equilibrium problems?
Start by writing the balanced equation and K expression, then make an ICE table (Initial, Change, Equilibrium): 1) List species across top and rows: Initial [or P], Change = ±x (use stoichiometric ratios), Equilibrium = initial ± x. 2) Fill Initial from the problem (mol/L or atm). If a reactant is completely absent, its initial = 0. 3) For Change, assign −coeff·x for reactants consumed and +coeff·x for products formed (keep stoichiometry consistent). 4) Express K (Kc or Kp) in terms of equilibrium concentrations/pressures and substitute the expressions from the ICE row. 5) Solve for x. If you get a quadratic, solve exactly or use the small-x approximation (if K is ≤10−3 or initial >> x, assume x small; check that x/initial < 5% afterwards). 6) Compute equilibrium values and report with correct units/significant figures. This is exactly what Topic 7.7 tests—practice setting up ICEs and using Q vs K to predict direction. For worked examples, see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and try problems at Fiveable’s practice bank (https://library.fiveable.me/practice/ap-chemistry).
What happens to the concentrations when Q is greater than K?
If Q > K the system has too many products relative to equilibrium, so the reaction will shift left (toward reactants) until Q decreases to equal K. Practically, that means there’s a net consumption of products and net formation of reactants—product concentrations fall and reactant concentrations rise—while the forward and reverse rates move toward equality at the new equilibrium. You can show this with an ICE table: put a negative change for products and a corresponding positive change for reactants (using stoichiometry) and solve for equilibrium concentrations. This is exactly Essential Knowledge 7.7.A.2 and is tested on the AP exam when you’re asked to predict direction or compute equilibrium concentrations. For extra worked examples and practice, see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and try problems from Fiveable’s AP practice set (https://library.fiveable.me/practice/ap-chemistry).
Can someone explain how to predict which direction a reaction will go using Q and K?
Compute Q the same way you write K (use concentrations for Kc or partial pressures for Kp) but plug in the initial/current values instead of equilibrium values. Then compare: - If Q < K → net reaction goes right (reactants consumed, products formed) until Q = K. - If Q > K → net reaction goes left (products consumed, reactants formed) until Q = K. - If Q = K → system is at dynamic equilibrium (rates forward = reverse). Quick how-to: write the equilibrium expression from the balanced equation (use stoichiometric exponents), substitute initial [ ] or P to get Q, then compare to the given K. If you need numbers, set up an ICE table and solve for the change (use quadratic approximation if x is small). This is exactly the 7.7.A skill on the AP CED—practice ICE/Q vs K problems to get fast and accurate. For worked examples and practice sets, see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO), the Unit 7 overview (https://library.fiveable.me/ap-chemistry/unit-7), and 1000+ practice problems (https://library.fiveable.me/practice/ap-chemistry).
I don't understand how to use the equilibrium constant to find unknown concentrations - can you walk me through it?
Start with the balanced equation and K (Kc or Kp). Make an ICE table: write Initial concentrations (or partial pressures), the Change (use an unknown x tied to stoichiometry), and the Equilibrium expressions (Initial ± change = Equilibrium). Write K = (products)^{coeff}/(reactants)^{coeff} using the equilibrium values from the table and solve for x. Quick checklist: - If you can, calculate Q first: Q < K → reaction makes products (x positive); Q > K → reaction makes reactants (x negative). (CED 7.7.A.2) - Substitute equilibrium values into K and solve algebraically. Often you get a quadratic—solve exactly or use the quadratic approximation (if K is very small and initial >> x, assume x ≪ initial; check that x/initial < 5%). - Convert partial pressures if using Kp (use Kp vs Kc relationship if needed). Example pattern for aA + bB ⇌ cC: I: [A]0, [B]0, 0 C: -c*x, -b*x, +a*x E: [A]0 - c x, [B]0 - b x, a x K = ([C]^{a})/([A]^{c}[B]^{b}) → solve for x. For AP prep, practice ICE/quadratic problems (Topic 7.7 study guide: https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO). For more mixed practice, see the Unit 7 overview (https://library.fiveable.me/ap-chemistry/unit-7) and thousands of practice problems (https://library.fiveable.me/practice/ap-chemistry).
What's the difference between using partial pressures and concentrations in equilibrium calculations?
Use concentrations (Kc) when you’re working with molarities [M], and use partial pressures (Kp) when the reacting species are gases and you’re given pressures. Both express the same equilibrium idea (Q < K → forward; Q > K → reverse) and you can solve equilibrium with an ICE table either way (CED 7.7.A). Key links between them: - Kp = Kc (RT)Δn, where Δn = moles gas products − moles gas reactants. So if you know Kc you can get Kp (and vice versa) using R and T. - You can convert a gas’s partial pressure to concentration with [A] = PA / (RT) if you need to use Kc instead of Kp. Also note: heterogeneous equilibria omit pure solids and liquids from the expression (only gases/aqueous appear). On the AP exam you’ll be asked to set up equilibrium expressions, compute Q or K (Kc vs Kp), and use ICE tables and the quadratic or approximation when needed. For a focused review see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO), the Unit 7 overview (https://library.fiveable.me/ap-chemistry/unit-7), and lots of practice problems (https://library.fiveable.me/practice/ap-chemistry).
How do I know if I should use Kc or Kp in my equilibrium problem?
Use Kc when the equilibrium expression is written with concentrations (mol·L⁻¹); use Kp when it’s written with partial pressures (atm). For gas-phase reactions you can use either, but convert with the relation Kp = Kc(RT)Δn, where Δn = (moles gas products) − (moles gas reactants), R = 0.08206 L·atm·mol⁻¹·K⁻¹, and T is in K. Heterogeneous equilibria: ignore pure solids/liquids in either Kc or Kp expressions (they don’t appear in the equilibrium expression). When solving problems, pick the K that matches the given equilibrium data (concentrations → Kc, pressures → Kp); if you get the other K, convert using the Kp–Kc equation. Also remember to compare Qc or Qp to K to predict direction (Q < K → forward, Q > K → reverse), which is tested in LO 7.7.A. For more worked examples, see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and practice problems (https://library.fiveable.me/practice/ap-chemistry).
Why do we say equilibrium is dynamic when the concentrations stay constant?
Saying equilibrium is dynamic means reactions keep happening, but the net change is zero because the forward and reverse rates are equal. Molecules still collide and convert between reactants and products, but as many moles form in the forward direction per second as are converted back per second, so concentrations stay constant. That’s exactly what Q = K tells you (Topic 7.3–7.7): when Q = K the system is at dynamic equilibrium (CED 7.7.A.2). Use an ICE table to see this numerically—the x that changes reactant and product concentrations cancels out so the final amounts don’t change, even though x represents ongoing reaction. If you disturb the system, Le Chatelier’s principle predicts a shift until forward and reverse rates equalize again. For more practice and worked examples on calculating equilibrium concentrations and seeing this in ICE tables, check the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) or the Unit 7 overview (https://library.fiveable.me/ap-chemistry/unit-7).
What are the steps to solve for equilibrium concentrations when given initial conditions?
1) Write the balanced equation and the K expression (Kc or Kp)—this is required by the CED (Topic 7.7). 2) List initial concentrations/pressures for all species. Calculate Q to see the reaction direction (Q < K → makes products; Q > K → makes reactants). 3) Build an ICE table (Initial, Change, Equilibrium). Use stoichiometric multipliers for the change row (±x, ±2x, etc.). 4) Substitute equilibrium values into the K expression and solve for x. That may give: a simple algebraic solution, or a quadratic. If you get a quadratic, solve exactly or use the quadratic approximation (assume x is small if K is ≪1 and initial >> x; check that x/initial < ~5%). 5) Compute equilibrium concentrations/partial pressures and check by plugging back into K. If a solid/liquid is present, omit its concentration from K (heterogeneous equilibrium). For step-by-step examples and AP-style practice, see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and try lots of practice problems (https://library.fiveable.me/practice/ap-chemistry).
How do you calculate Q and compare it to K to predict reaction direction?
Calculate Q the same way as K but using the current (not necessarily equilibrium) concentrations or partial pressures: write the equilibrium expression from the balanced reaction (products^coefficients / reactants^coefficients) and plug in the initial or current values to get Q (Kc or Kp form). Then compare Q to K: - If Q < K → system shifts right (net forward reaction): reactants consumed, products formed. - If Q > K → system shifts left (net reverse reaction): products consumed, reactants formed. - If Q = K → system is at dynamic equilibrium. Quick steps: (1) write balanced equation and K expression, (2) plug initial concentrations to get Q, (3) compare with given K and state direction. Use an ICE table when you need equilibrium concentrations after predicting the shift; apply quadratic approximation if needed (Topic 7.7 guidance). For a refresher see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and lots of practice problems at (https://library.fiveable.me/practice/ap-chemistry).
I missed the lab on equilibrium - how do you actually measure when a system reaches equilibrium?
In the lab you judge equilibrium by measuring concentrations (or partial pressures) over time until they stop changing—that steady state means Q = K and forward/reverse rates are equal. Practically you pick a measurable property that tracks a species: common methods are UV–Vis or colorimetry (absorbance vs concentration), titration of aliquots to find [acid/base] or [ion], pressure measurement for gaseous reactions (manometer), conductivity for ions, or spectroscopic peaks for specific species. Take repeat measurements at intervals, plot concentration vs time, and when values level off (within experimental uncertainty) you’re at equilibrium. Use an ICE table and the known K to check: compute Q from your measured concentrations—when Q ≈ K you’re done. For AP-style calculations practice setting up ICE tables, using Kc/Kp, and checking approximations (quadratic)—see the Topic 7.7 study guide (https://library.fiveable.me/ap-chemistry/unit-7/calculating-equilibrium-concentrations/study-guide/ou9xNlxg758Auz6D3WIO) and the Unit 7 overview (https://library.fiveable.me/ap-chemistry/unit-7). For extra practice, try problems at (https://library.fiveable.me/practice/ap-chemistry).