Calculate pH of 10 M Cr(NO3)3
This premium calculator estimates the pH of an aqueous chromium(III) nitrate solution by treating the hydrated chromium(III) ion as a weak acid. For a typical classroom approximation, Cr(NO3)3 fully dissociates, nitrate acts as a spectator ion, and acidity comes from hydrolysis of the hexaaquachromium(III) complex.
Interactive pH Calculator
Expert Guide: How to Calculate the pH of 10 M Cr(NO3)3
When students search for how to calculate the pH of 10 M Cr(NO3)3, they are usually asking a deceptively simple acid-base question that actually sits at the intersection of coordination chemistry, hydrolysis, and approximation methods. Chromium(III) nitrate, written as Cr(NO3)3, is a soluble ionic compound. In water it dissociates to chromium(III) ions and nitrate ions. Because nitrate is the conjugate base of a strong acid, nitric acid, it contributes essentially no basicity in introductory pH calculations. The interesting chemistry comes from the Cr3+ ion, which strongly polarizes the coordinated water molecules in its hydration shell. That effect makes the hydrated ion acidic.
In practical general chemistry, the chromium(III) ion is often represented as the complex ion [Cr(H2O)6]3+. This hydrated species behaves as a weak acid according to the equilibrium:
[Cr(H2O)6]3+ + H2O ⇌ [Cr(H2O)5OH]2+ + H3O+
Because hydronium is produced, the solution becomes acidic. Therefore, to estimate the pH of 10 M Cr(NO3)3, you usually do not assume the salt is neutral. Instead, you model the hydrated chromium(III) cation as a weak acid with a pKa often taken around 4.0 for classroom-level work. Once you have a pKa, the problem becomes a standard weak-acid equilibrium calculation.
Short answer for 10 M Cr(NO3)3
If you use a typical value of pKa = 4.00 for the hydrated chromium(III) ion, then:
- Ka = 10-4 = 0.0001
- Initial acid concentration, C = 10.0 M
- For a weak acid, x ≈ √(Ka × C) = √(0.0001 × 10) = √0.001 ≈ 0.0316 M
- pH = -log(0.0316) ≈ 1.50
Using the exact quadratic solution instead of the square-root approximation gives essentially the same answer at this precision: pH ≈ 1.50.
Step-by-step method
- Write the dissociation of the salt: Cr(NO3)3 → Cr3+ + 3 NO3-
- Identify the acidic species: NO3- is negligible in acid-base behavior, but Cr3+ hydrolyzes water.
- Represent the metal ion as a weak acid: [Cr(H2O)6]3+ ⇌ [Cr(H2O)5OH]2+ + H+
- Choose the acid constant: a common instructional value is pKa = 4.00, so Ka = 1.0 × 10-4.
- Set up an ICE table: initial concentration of acid is 10.0 M, change is -x, and equilibrium concentrations are 10.0 – x, x, and x.
- Use the Ka expression: Ka = x2 / (10.0 – x)
- Solve for x: x is the hydronium concentration.
- Find pH: pH = -log[H3O+]
Exact weak-acid calculation
For greater rigor, solve the quadratic rather than using the approximation. Starting from:
Ka = x2 / (C – x)
Rearrange:
x2 + Ka x – Ka C = 0
Then:
x = (-Ka + √(Ka2 + 4KaC)) / 2
Substituting Ka = 1.0 × 10-4 and C = 10.0:
x = (-0.0001 + √(0.00000001 + 0.004)) / 2 ≈ 0.03157 M
Finally:
pH = -log(0.03157) ≈ 1.50
Why the nitrate ion does not control the pH
Many salt pH problems can be solved by checking whether the ions come from strong or weak acids and bases. In this case, nitrate is the conjugate base of nitric acid, a strong acid, so nitrate itself has extremely weak basicity. That means it does not appreciably consume protons from water. The entire acidity of the solution is dominated by the small but chemically meaningful hydrolysis of the highly charged chromium(III) ion.
The reason Cr3+ is acidic is charge density. Small, highly charged metal ions strongly attract the oxygen atoms of coordinated water molecules, withdrawing electron density from O-H bonds. That makes proton release easier than in free water. This is a central concept in aqueous coordination chemistry and explains why many multivalent metal cations form acidic solutions.
Important realism check for a 10 M solution
A 10 M chromium(III) nitrate solution is an extremely concentrated solution. At concentrations this high, a simple textbook model has limitations. Real solutions can depart from ideality because of:
- Ionic strength effects: activities differ from concentrations.
- Complex speciation: multiple hydrolyzed chromium species may exist.
- Hydration structure changes: concentrated electrolyte environments alter equilibrium behavior.
- Temperature dependence: hydrolysis constants can shift with temperature.
So the answer of about pH 1.50 should be understood as a solid educational estimate, not a full thermodynamic model for a highly concentrated industrial solution. In laboratory research or process design, activity corrections and species distribution models would be more appropriate.
Comparison table: approximate pH across concentration
| Cr(NO3)3 concentration | Assumed pKa of [Cr(H2O)6]3+ | Estimated [H+] | Estimated pH | Percent hydrolyzed |
|---|---|---|---|---|
| 0.010 M | 4.00 | 0.00100 M | 3.00 | 10.00% |
| 0.100 M | 4.00 | 0.00316 M | 2.50 | 3.16% |
| 1.00 M | 4.00 | 0.0100 M | 2.00 | 1.00% |
| 5.00 M | 4.00 | 0.0224 M | 1.65 | 0.45% |
| 10.00 M | 4.00 | 0.0316 M | 1.50 | 0.32% |
This table highlights a subtle point: as concentration rises, pH drops, but the fraction hydrolyzed actually becomes smaller. That behavior is typical of weak acids. A more concentrated weak acid gives more absolute hydronium, yet a smaller percentage of the acid molecules ionize.
Comparison with other common ions
To better understand chromium(III), it helps to compare it with ions students often classify in introductory chemistry:
| Ion in water | Acid-base behavior | Main reason | Typical classroom pH effect |
|---|---|---|---|
| Na+ | Essentially neutral | Very weak interaction with water acidity | Near no change |
| NH4+ | Weakly acidic | Conjugate acid of weak base NH3 | Mild pH decrease |
| Al3+ | Acidic | High charge density and hydrolysis | Strong pH decrease |
| Cr3+ | Acidic | Hydrolysis of hydrated metal ion | Strong pH decrease |
| NO3- | Essentially neutral | Conjugate base of strong acid HNO3 | Near no change |
Common mistakes to avoid
- Calling Cr(NO3)3 neutral: this ignores hydrolysis of Cr3+.
- Treating Cr3+ as a strong acid: it is acidic, but not fully dissociated in the Brønsted sense used for strong monoprotic acids.
- Using nitrate in the equilibrium expression: nitrate does not drive the pH here.
- Ignoring concentration limits: 10 M is so concentrated that ideal approximations become less reliable.
- Confusing molarity with acidity strength: a high concentration of a weak acid can still have significant acidity.
When the square-root approximation works
The approximation x ≈ √(KaC) works when x is small relative to the starting concentration C. For the 10 M example:
- x ≈ 0.0316 M
- x/C ≈ 0.00316, or 0.316%
That is comfortably under the typical 5% rule, so the approximation is justified. In fact, this is why the approximate and exact answers are nearly identical for this case.
Interpretation of the result
A pH near 1.50 means the solution is strongly acidic by practical standards, even though the acidity originates from hydrolysis of a metal aqua complex rather than from direct addition of a strong acid such as HCl or HNO3. That is an important conceptual lesson: metal salts are not automatically neutral in water. The identity and charge of the cation matter enormously.
If you are solving a homework or exam problem and no Ka or pKa is given, your instructor may expect one of two approaches: either assume the hydrated chromium(III) ion is a weak acid with a provided or memorized pKa, or state that a numerical pH cannot be calculated exactly without a hydrolysis constant. If the problem explicitly asks for the pH of 10 M Cr(NO3)3 and expects a number, the common textbook-style answer is generally based on a pKa around 4, giving pH ≈ 1.50.
Recommended authoritative references
- LibreTexts Chemistry for hydrolysis and metal ion acidity explanations.
- NIST Chemistry WebBook for reliable chemical data and reference material.
- U.S. Environmental Protection Agency for chromium chemistry and environmental context.
- U.S. Geological Survey for chromium occurrence and water chemistry context.
Final takeaway
To calculate the pH of 10 M Cr(NO3)3, treat chromium(III) as an acidic hydrated ion and nitrate as a spectator ion. Using a typical classroom value of pKa = 4.00 for [Cr(H2O)6]3+, the estimated hydrogen ion concentration is about 3.16 × 10-2 M, which gives pH ≈ 1.50. That is the standard educational result, while also remembering that very concentrated real solutions may require more advanced activity-based modeling.
Educational note: this page provides an estimation model suitable for classroom chemistry and quick calculations. Highly concentrated solutions can show non-ideal behavior beyond the scope of a simple weak-acid treatment.