Calculate the pH of a 0.108 m Solution of CsHSO4
Use this premium chemistry calculator to estimate the pH of cesium hydrogen sulfate solution by modeling HSO4− as a weak acid in water. The tool shows pH, hydronium concentration, percent dissociation, and a concentration chart for quick interpretation.
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How to calculate the pH of a 0.108 m solution of CsHSO4
To calculate the pH of a 0.108 m solution of CsHSO4, you first identify the chemistry of the dissolved species. CsHSO4 is cesium hydrogen sulfate. In water, the cesium ion, Cs+, behaves as a spectator ion because it comes from the strong base CsOH and does not significantly hydrolyze under ordinary conditions. The chemically important species for acid-base behavior is HSO4−, the hydrogen sulfate ion.
Hydrogen sulfate is amphiprotic, but in this context it is most commonly treated as an acid because it can donate a proton to water:
The acid dissociation constant for this step is typically taken as about Ka = 1.2 × 10-2 at room temperature. That means HSO4− is not a strong acid like the first dissociation of sulfuric acid, but it is still substantially acidic. Because the given concentration is 0.108 m, and because dilute aqueous molality and molarity are often numerically close, many textbook and homework solutions approximate 0.108 m as 0.108 M for pH estimation. This calculator follows that conventional approach unless you specify otherwise.
Step-by-step setup
Start with an initial concentration of hydrogen sulfate equal to the concentration of dissolved cesium hydrogen sulfate:
Let x be the amount of HSO4− that dissociates:
- [H3O+] = x
- [SO4^2−] = x
- [HSO4−] = 0.108 – x
Substitute these into the equilibrium expression:
Using Ka = 0.012:
Now solve the quadratic equation:
The physically meaningful root is approximately:
Since x equals the hydronium concentration, the pH is:
So the pH of a 0.108 m solution of CsHSO4 is approximately 1.52 when treated using the standard Ka for HSO4− and a dilute-solution approximation.
Why CsHSO4 is acidic
Many students initially focus on the metal cation and wonder whether cesium contributes to the pH. In reality, Cs+ is a Group 1 alkali metal ion and is a very weak Lewis acid in water. It does not noticeably acidify the solution. The acidity comes from the hydrogen sulfate ion, which still has one acidic proton available to donate.
This is an important distinction in salt hydrolysis. Not every salt is neutral. A salt formed from a strong base and a polyprotic acid can still be acidic if the anion retains an ionizable proton. CsHSO4 is exactly such a case. It is therefore more accurate to think of the dissolved solution as containing HSO4− ions that partially dissociate rather than a neutral salt that simply disperses in water.
Why we use the second dissociation of sulfuric acid
Sulfuric acid, H2SO4, dissociates in two stages:
- H2SO4 → H+ + HSO4− (essentially complete in water)
- HSO4− ⇌ H+ + SO4^2− (partial, governed by Ka)
Because CsHSO4 already contains HSO4−, you are working directly with the second dissociation step. That is why the relevant equilibrium constant is the Ka for hydrogen sulfate, not the much stronger first dissociation of sulfuric acid.
Data table: relevant acid-base constants and properties
| Species / property | Typical value at about 25°C | Why it matters here |
|---|---|---|
| Ka for HSO4− | 1.2 × 10-2 | Controls the equilibrium that generates H3O+ |
| pKa for HSO4− | 1.92 | Shows hydrogen sulfate is a fairly strong weak acid |
| Kw for water | 1.0 × 10-14 | Water autoionization is negligible compared with acid contribution |
| Initial CsHSO4 concentration | 0.108 | Starting concentration of HSO4− in the solution model |
| Calculated [H3O+] | ≈ 0.0305 | Directly used to compute pH |
Approximation versus quadratic solution
In weak-acid problems, students are often taught the shortcut:
If you apply that shortcut here, you obtain:
That gives a pH of about 1.44, which is somewhat lower than the more exact answer of 1.52. The reason the approximation is less accurate here is that dissociation is not tiny compared with the initial concentration. In fact, x is a fairly large fraction of 0.108, so subtracting x in the denominator matters. This is exactly the kind of problem where the quadratic method is preferred.
As a rule of thumb, if the percent dissociation is more than about 5%, the weak-acid approximation may introduce noticeable error. For 0.108 concentration hydrogen sulfate, the dissociation is much larger than that threshold, so a quadratic solution is the professional choice.
Comparison table: exact and approximate results
| Method | [H3O+] | pH | Percent dissociation |
|---|---|---|---|
| Quadratic equilibrium solution | 0.0305 | 1.52 | 28.2% |
| Weak-acid approximation | 0.0360 | 1.44 | 33.3% |
| Difference | 0.0055 | 0.08 pH unit | 5.1 percentage points |
Interpreting the result
A pH of about 1.52 means the solution is strongly acidic on the everyday pH scale, even though the acidic species is technically a weak acid in the equilibrium sense. This apparent contradiction is common in chemistry. A weak acid can still produce a very low pH if the concentration is high enough and the Ka is sufficiently large. Hydrogen sulfate is a classic example because its Ka is much larger than that of acetic acid, formic acid, or many common weak acids encountered in introductory chemistry.
At pH 1.52, the hydronium concentration is about 0.0305 mol/L, which is substantial. This means the solution should be handled using standard laboratory acid precautions, including splash protection and appropriate gloves, depending on the exact experimental setup and concentration basis.
What about molality versus molarity?
The problem statement uses 0.108 m, which formally means molality. pH equations are usually written in terms of activities and often approximated using molarity or effective concentration. In many classroom settings, a dilute aqueous solution at this level is treated as though 0.108 m is close enough to 0.108 M for a practical pH estimate. A more rigorous treatment would require density data and activity corrections. Those refinements may shift the final value slightly, but for standard general chemistry work, pH ≈ 1.52 is the accepted result.
Common mistakes to avoid
- Treating CsHSO4 as neutral: It is not neutral because HSO4− can still donate a proton.
- Using the first dissociation of sulfuric acid: That applies to H2SO4, not directly to hydrogen sulfate salts.
- Ignoring the quadratic requirement: The approximation is weaker here because dissociation is not small.
- Confusing m and M: In strict thermodynamic work, they are different. In many textbook calculations, they are treated as similar at low to moderate concentration.
- Forgetting that pH depends on hydronium: You must solve for [H3O+] before taking the negative logarithm.
Practical chemistry insight
If you compare hydrogen sulfate with weaker acids, its behavior stands out. Acetic acid has a Ka of about 1.8 × 10-5, while hydrogen sulfate is around 1.2 × 10-2. That means HSO4− is roughly hundreds of times more prone to donate a proton than acetic acid under comparable conditions. As a result, salts that contain HSO4− can produce surprisingly acidic solutions.
This is also why acid salts must be interpreted carefully in laboratory calculations. Compounds such as NaHSO4, KHSO4, and CsHSO4 all contain the hydrogen sulfate ion and therefore generate acidic aqueous solutions. The exact pH depends mostly on concentration and the hydrogen sulfate equilibrium, not on the identity of the alkali metal cation.
Reference concentrations and expected pH trend
As concentration increases, pH decreases because more hydrogen sulfate is available to dissociate. The relationship is not perfectly linear because equilibrium moderates the amount of hydronium formed. Still, the trend is clear: higher concentration means lower pH.
| HSO4− concentration | Approximate exact [H3O+] | Approximate pH |
|---|---|---|
| 0.010 | 0.0050 | 2.30 |
| 0.050 | 0.0190 | 1.72 |
| 0.108 | 0.0305 | 1.52 |
| 0.200 | 0.0437 | 1.36 |
Authoritative sources for deeper study
If you want to review the underlying acid-base theory, chemical safety, and aqueous equilibrium concepts in more depth, these authoritative resources are excellent places to start:
- LibreTexts Chemistry for acid-base equilibrium tutorials and worked examples.
- NIST Chemistry WebBook for reliable chemical reference information from a U.S. government source.
- U.S. Environmental Protection Agency for pH fundamentals and water chemistry context.
Bottom line
To calculate the pH of a 0.108 m solution of CsHSO4, model the dissolved salt as providing 0.108 concentration of HSO4−, use the hydrogen sulfate acid dissociation constant, and solve the equilibrium expression. Because the dissociation is significant, the quadratic equation gives the best answer. That yields a hydronium concentration of about 0.0305 and a pH of approximately 1.52.