Calculate pH of 10 M Cr(NO3)3
Use this interactive chromium(III) nitrate pH calculator to estimate the acidity of an aqueous Cr(NO3)3 solution. The model treats nitrate as a spectator ion and chromium(III) as an acidic hydrated metal ion, then solves for hydronium concentration using the acid dissociation constant of the hexaaquachromium(III) complex.
Expected pH range
1.500
[H+] estimate
3.152e-2 M
How to calculate the pH of 10 M Cr(NO3)3 correctly
If you need to calculate the pH of 10 M Cr(NO3)3, the key idea is that chromium(III) nitrate behaves as an acidic salt in water. Many students initially look at the nitrate ion and wonder whether nitrate affects the answer. In standard aqueous acid-base treatment, it does not. Nitrate, NO3-, is the conjugate base of a strong acid, HNO3, so it is essentially pH-neutral in this context. The acidity comes from the metal ion: Cr3+ exists in water as a hydrated complex, usually written as [Cr(H2O)6]3+. That metal aqua ion can donate a proton to water, which produces hydronium and lowers the pH.
A practical way to solve the problem is to treat hydrated chromium(III) as a weak acid with an acid dissociation constant Ka near 1.0 × 10-4. This is the assumption used in many general and analytical chemistry settings when a more detailed speciation model is not required. The calculator above uses that framework and solves either by the full quadratic equation or by the weak-acid approximation. For a concentration as high as 10.0 M, the full quadratic solution is the better of the two simple choices, even though the solution is concentrated enough that non-ideal behavior may matter in the laboratory.
Step 1: Write the acid hydrolysis equilibrium
The relevant Brønsted acid reaction is the first hydrolysis of the chromium(III) aqua complex:
Here, the chromium complex donates a proton from a coordinated water ligand. This is why metal ions with high positive charge and relatively small size often acidify water. Chromium(III) strongly attracts electron density from coordinated water molecules, weakening the O-H bonds enough to make proton loss more favorable.
Step 2: Set up the equilibrium expression
Let the initial formal concentration of Cr(NO3)3 be C = 10.0 M. If x moles per liter of hydronium are produced by hydrolysis, then the standard ICE setup gives:
- Initial [Cr(H2O)6]3+ = 10.0 M
- Change = -x
- Equilibrium [Cr(H2O)6]3+ = 10.0 – x
- Equilibrium [H3O+] = x
- Equilibrium [[Cr(H2O)5OH]2+] = x
The equilibrium expression is therefore:
Using Ka = 1.0 × 10-4:
Step 3: Solve for hydronium concentration
Rearranging gives the quadratic:
With C = 10.0 and Ka = 1.0 × 10-4:
Since x represents [H3O+], we have:
So the standard textbook estimate for the pH of 10 M Cr(NO3)3 is about 1.50.
Why nitrate does not control the pH
It is helpful to say this clearly because it is a common source of mistakes. Chromium(III) nitrate contains three nitrate ions, but nitrate is not a significant base in water. Since nitric acid is a strong acid, its conjugate base is very weak and does not materially consume hydronium under normal conditions. In other words, the salt is acidic because of Cr3+, not because nitrate somehow becomes acidic on its own.
This is a general pattern in inorganic chemistry:
- Highly charged metal cations often make water acidic through hydrolysis.
- Anions from strong acids, such as nitrate, chloride, and perchlorate, are usually spectators for pH calculations.
- The stronger the metal ion’s hydrolyzing tendency, the lower the pH of its solution at the same concentration.
Approximation versus exact solution
In many classes, you may also see the weak-acid approximation:
Plugging in the same values:
The approximation happens to work very well here because x is still small relative to 10.0 M. However, for polished work, the quadratic solution is preferable and is what the calculator uses by default.
Comparison table: estimated pH versus concentration
The acidity changes with concentration. Using the same idealized Ka value of 1.0 × 10-4, the table below shows how the estimated pH varies as the formal concentration changes.
| Cr(NO3)3 concentration (M) | Estimated [H+] (M) | Estimated pH | Interpretation |
|---|---|---|---|
| 0.001 | 3.11 × 10-4 | 3.507 | Mildly acidic dilute solution |
| 0.01 | 9.95 × 10-4 | 3.002 | Clear hydrolysis effect |
| 0.10 | 3.11 × 10-3 | 2.507 | Stronger acidity from greater formal concentration |
| 1.00 | 9.95 × 10-3 | 2.002 | Strongly acidic metal salt solution |
| 10.0 | 3.16 × 10-2 | 1.501 | Very acidic, very concentrated solution |
What makes 10 M special and why the estimate has limits
Although the equilibrium setup above is standard and chemically meaningful, 10 M is a very high concentration. At that level, the solution’s ionic strength is enormous, and activities are no longer equal to concentrations. In rigorous physical chemistry, pH is related to the activity of hydronium rather than its simple molar concentration. That means a measured pH in a real 10 M chromium(III) nitrate solution can differ from the textbook estimate. In addition, highly concentrated metal salt solutions may show complex hydration behavior, multiple hydrolysis steps, and deviations in apparent equilibrium constants.
For exam work, homework, and conceptual problem-solving, however, the idealized weak-acid model remains the accepted approach unless the problem explicitly asks for activity corrections or advanced speciation. So if your instructor says “calculate the pH of 10 M Cr(NO3)3,” the most defensible standard answer is still about pH = 1.50.
Composition table for a 10.0 M Cr(NO3)3 solution
Another useful perspective is to see how much dissolved material is present. The molar mass of anhydrous Cr(NO3)3 is approximately 238.01 g/mol, so a 10.0 M solution corresponds to an extremely large dissolved mass per liter.
| Quantity | Value for 10.0 M Cr(NO3)3 | How it is obtained |
|---|---|---|
| Molar mass of Cr(NO3)3 | 238.01 g/mol | 51.996 + 3 × 62.004 |
| Total dissolved Cr(NO3)3 per liter | 2380.1 g/L | 10.0 mol/L × 238.01 g/mol |
| Chromium content per liter | 519.96 g/L | 10.0 mol/L × 51.996 g/mol |
| Nitrate content per liter | 1860.1 g/L | 10.0 mol/L × 3 × 62.004 g/mol |
| Estimated hydronium from first hydrolysis | 0.0316 mol/L | Quadratic solution with Ka = 1.0 × 10-4 |
Regulatory perspective and real-world scale
Chemistry students often benefit from comparing calculated concentrations with familiar water-quality benchmarks. Drinking water standards are designed for trace levels, not the concentrated reagent conditions used in pH exercises. Still, the comparison helps show just how far a 10 M solution sits from environmental concentrations.
| Species or benchmark | Typical regulatory number | 10.0 M Cr(NO3)3 equivalent | Scale comparison |
|---|---|---|---|
| EPA drinking water MCL for total chromium | 0.1 mg/L | 519,961 mg/L chromium | About 5.2 million times higher |
| EPA MCL for nitrate as nitrogen | 10 mg/L as N | 420,201 mg/L as nitrate-nitrogen equivalent | About 42,000 times higher |
| EPA MCL for nitrate as nitrate | 45 mg/L as NO3- | 1,860,111 mg/L nitrate | About 41,300 times higher |
Common mistakes when solving this problem
- Assuming the salt is neutral: Cr(NO3)3 is not like NaNO3. Chromium(III) hydrolyzes and makes the solution acidic.
- Using nitrate as the acid source: nitrate is a spectator in this pH calculation.
- Ignoring the hydrated metal ion: the actual acid is [Cr(H2O)6]3+.
- Forgetting the quadratic option: even when approximation works, it is good practice to verify with the exact expression.
- Overstating precision: at 10 M, the idealized model is approximate because activity effects are important.
Best final answer for most coursework
Under the standard weak-acid hydrolysis model, taking Ka for hydrated Cr3+ as 1.0 × 10-4, the hydronium concentration in a 10.0 M Cr(NO3)3 solution is about 3.16 × 10-2 M. Therefore:
That is the answer most instructors expect unless your course has specifically covered activity coefficients, advanced metal-ion speciation, or concentration-dependent equilibrium constants. If you are preparing content for a lab or a technical report, it is wise to describe the result as an idealized estimate rather than an exact measured pH.
Authoritative references and further reading
For broader context on chromium chemistry, pH measurement, and water-quality benchmarks, see these authoritative sources: