pH Calculator with Steps: pH, pOH, Ka, Buffer | StatsUnlock

To calculate pH, take the negative base-10 logarithm of the hydrogen ion concentration: pH = −log[H⁺]. For a strong acid, [H⁺] is its molarity, so 0.010 M HCl has pH 2.00. For a weak acid, first find [H⁺] from Ka: for 0.10 M acetic acid (Ka = 1.8 × 10⁻⁵),

Rule of thumb: pH + pOH = 14.00 at 25 °C, and each pH unit is a tenfold change in [H⁺]. Below 7 is acidic, 7 is neutral and above 7 is basic.

pH is a measure of how acidic or basic a water solution is. It is defined as the negative logarithm of the hydrogen ion activity, which in dilute solutions is close to the concentration in mol/L: pH = −log₁₀[H⁺] (Buck et al., 2002). Because the scale is logarithmic, a solution at pH 3 has ten times more H⁺ than one at pH 4 and a thousand times more than one at pH 6.

A pH calculator turns what you know about a solution into [H⁺] and then into pH. For a strong acid that is one step; for a weak acid it means solving the equilibrium with Ka; for a buffer it uses the ratio of base to acid; during a titration it tracks how many moles are left. This calculator solves every case from the full charge balance, including the H⁺ and OH⁻ that water itself supplies, and shows the textbook shortcut next to the exact answer so you can see when the shortcut is good enough.

A worked one-liner: [H⁺] = 2.5 × 10⁻⁴ M gives pH = −log(2.5 × 10⁻⁴) = 3.60 and pOH = 14.00 − 3.60 = 10.40.

Who uses it: pH calculations are taught in NCERT Class 11 (“Equilibrium”) and asked every year in JEE and NEET; GCSE, A level, AP Chemistry and IB courses cover pH, pOH, Ka and buffers, and the POGIL activity “Calculating pH” is widely used in US classrooms. Biologists prepare buffers, analysts run titrations, and pool owners adjust their water: the same equations serve all of them.

What this free pH calculator solves

One tool covers every pH calculation in a typical course and a few beyond it. It works as a pH and pOH calculator, a pH concentration calculator for [H⁺] and [OH⁻], a pH molarity calculator for strong acids and bases, a weak acid pH calculator with Ka or pKa, a buffer pH calculator, a Ka-from-pH calculator, a titration pH calculator and a swimming pool pH calculator. Press Try it to load an example, then press the button to see the working.

Free, with no sign-up. Everything runs in your browser: nothing you type is sent to a server or stored.

En español (pH calcul): ¿cómo se calcula el pH? pH = −log[H⁺]. Para HCl 0,010 M, [H⁺] = 0,010 M y el pH es 2,00; para un ácido débil, [H⁺] = √(Ka × C).

How accurate is this pH calculator?

Every equilibrium answer is solved from the charge balance, [H⁺] + [cations] = [OH⁻] + [anions], with Kw = 1.0 × 10⁻¹⁴ at 25 °C (or a temperature-dependent Kw if you enter a temperature). That is exact for ideal dilute solutions, so it stays correct where shortcuts fail: very dilute acids, weak acids that are more than 5% ionised, and the equivalence point of a titration. pH is given to as many decimal places as the concentration has significant figures. Every answer is checked before it gets the Verified badge: pH + pOH must equal pKw, and Ka must come back out of the equilibrium concentrations. The R and Python scripts in Sections 10 and 11 return the same pH 2.87 (shortcut) and 2.88 (quadratic) for 0.10 M acetic acid.

Studies of how students learn pH find the logarithm itself is the main stumbling block: many can press the log key but cannot say what a change of one pH unit means (Watters & Watters, 2006). That is why the results show [H⁺] and pH side by side and plot them on a log scale.

4.1 Acidic, neutral and basic

At 25 °C a solution with pH below 7 is acidic ([H⁺] > [OH⁻]), pH 7 is neutral and above 7 is basic. Neutral means [H⁺] = [OH⁻], not pH 7 everywhere: Kw grows with temperature, so neutral water is pH 7.47 at 0 °C and about 6.63 at 50 °C. Enter a temperature in the calculator to see this.

4.2 Reading the logarithm

A change of one pH unit is a factor of ten in [H⁺]; two units is a factor of one hundred. Lemon juice (pH about 2) is 100 000 times more acidic than pure water (pH 7). pH can be below 0 or above 14 for very concentrated acids and bases, but there the simple formula, which uses concentration instead of activity, stops being accurate.

4.3 Significant figures in pH

The digits before the decimal point in a pH only set the power of ten; the decimal places carry the precision. So a pH has as many decimal places as the concentration has significant figures: 0.010 M (two significant figures) gives pH 2.00, and pH 4.20 gives [H⁺] = 6.3 × 10⁻⁵ M (two significant figures).

4.4 Strong versus weak

A strong acid gives up all its H⁺; a weak acid gives up only a small fraction. At 0.10 M, HCl has pH 1.00 while acetic acid has pH 2.88, because only 1.3% of the acetic acid molecules are ionised. Diluting a weak acid raises the fraction that ionises, which is why percent ionisation rises in chart 4 of the weak-acid results.

4.5 When the √(Ka C) shortcut is safe: the 5% rule

The shortcut assumes x is so small that C − x ≈ C. Check it: if x ÷ C × 100 is under 5%, the shortcut and the quadratic agree to about 0.01 pH. For 0.010 M HF (Ka = 6.8 × 10⁻⁴) the shortcut gives x = 26% of C, so it fails: the shortcut says pH 2.58, the quadratic 2.64. The calculator flags this automatically.

4.6 Buffers and the Henderson–Hasselbalch equation

A buffer contains a weak acid and its conjugate base. Added acid is taken up by A⁻ and added base by HA, so the pH barely moves. pH = pKa + log([A⁻] ÷ [HA]) works well when both concentrations are much larger than [H⁺] and the ratio lies between 0.1 and 10 (Po & Senozan, 2001). The buffer capacity tells you how much strong acid or base it can absorb per pH unit; it is highest at pH = pKa.

4.7 Titration curves and equivalence points

At the equivalence point of a strong acid–strong base titration the pH is 7.00. For a weak acid titrated with NaOH it is above 7 (acetic acid: 8.73 at 0.0500 M acetate) because the conjugate base reacts with water. Halfway to equivalence [HA] = [A⁻], so pH = pKa, the usual way to measure a pKa in the lab. Choose an indicator that changes colour inside the steep jump around equivalence.

4.8 A field note on where these problems go wrong

In marking pH work, the same slips come up again and again: using the natural log instead of log₁₀, forgetting to double [H⁺] for H₂SO₄ or [OH⁻] for Ba(OH)₂, giving a base a pH below 7 by reporting pOH as pH, and treating a weak acid as strong. Research on buffers finds many students cannot say which species reacts with added acid (Orgill & Sutherland, 2008). The calculator shows every intermediate value so each slip is easy to spot.

▶ Solve a problem above to fill all five cards with your own answer.

The formula to calculate pH is pH = −log[H⁺]. Every other pH equation below either finds [H⁺] or rearranges this one. Each card has a worked number.

This flowchart shows how to calculate the pH of a weak acid, on the same problem used in the Excel, R, Python and by-hand sections: calculate the pH of 0.10 M acetic acid, Ka = 1.8 × 10⁻⁵.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return 2.87 (shortcut) and 2.88 (quadratic) for this problem.

1. Choose what you know. Pick a strong acid or base, a pH or concentration to convert, a weak acid or base, a buffer, a measured pH, a titration point or a pool.
2. For a strong acid or base, pick it and type the molarity. H2SO4 and Ba(OH)2 count two H+ or OH-; choose another strong acid to set the number yourself.
3. To convert, choose pH, pOH, [H+] or [OH-]. Type the value; concentrations can be written as 2.5e-4.
4. For a weak acid or base, enter C and Ka, Kb or pKa. Or pick a compound such as acetic acid, HF or ammonia to fill in its typical value.
5. For a buffer, enter the pKa and both amounts. Use concentrations, or moles with the total volume; add a target pH to get the ratio you need.
6. For a titration, describe the flask and the burette. Enter concentrations, the flask volume, the volume added and the pKa for a weak acid or base.
7. For a pool, enter the volume, both pH values and the alkalinity. Add cyanuric acid if you use stabilised chlorine, and choose the acid product.
8. Press the button. Nothing is calculated before you press it, and changing any input clears the old answer.
9. Read the steps, the warnings and the charts. The Verified badge means pH + pOH = pKw and the equilibrium is satisfied.
10. Copy, share or batch. Copy the answer, share a link, download a report, or solve a whole worksheet on the Batch tab.

Excel calculates pH with one function, LOG10. For a weak acid add two cells, one for the square-root shortcut and one for the quadratic. The sheet below uses the headline problem, so it must return pH 2.87 (shortcut) and 2.88 (quadratic) for 0.10 M acetic acid, the same as the pH calculator above.

9.1 The functions you need

9.2 Step by step

9.3 Awkward cases

• Strong bases. pOH = =-LOG10(n*C), then pH = =14-pOH.
• Very dilute acids. Below 10⁻⁶ M use =-LOG10((C+SQRT(C^2+4E-14))/2) so water is included.
• Titration curves. Put volumes in column A and the charge-balance pH in column B; a scatter chart gives the curve. Goal Seek can solve the charge balance cell by cell.

9.4 Why your Excel answer might differ

9.5 Charting in Excel

1. Make a column of concentrations (1, 0.1, 0.01 and so on) and a column of pH values.
2. Insert → Scatter with Smooth Lines, then set the horizontal axis to a logarithmic scale, as in chart 4 of the weak-acid results.

9.6 Excel errors you will probably hit

This base R script calculates the pH of a weak acid three ways (the √(Ka C) shortcut, the quadratic and the full charge balance with uniroot), applies the 5% rule, checks seven textbook answers and saves a 300 dpi chart of weak and strong acid pH against concentration. For the headline problem it prints pH 2.87 and 2.88, matching the pH calculator on this page.

10.1 The complete script

R, base only, saves ph_curve.png

# ============================================================
# pH calculator in R (base R only): weak acid pH by the shortcut,
# the quadratic and the exact charge balance (water included)
# StatsUnlock - https://statsunlock.com/ph-calculator/
# ============================================================

# ---- 1. Your solution -------------------------------------------------
C  <- 0.10        # mol/L of the weak acid
Ka <- 1.8e-5      # acid dissociation constant (Ka <- 10^-pKa if you have pKa)
Kw <- 1.0e-14     # ion product of water at 25 degrees C

ph <- function(h) -log10(h)

ph_strong_acid <- function(c, n = 1) {        # water included
  a <- n * c
  ph((a + sqrt(a^2 + 4 * Kw)) / 2)
}

ph_weak_acid_exact <- function(c, ka) {       # charge balance, solved with uniroot
  f <- function(logh) { h <- 10^logh; h - ka * c / (ka + h) - Kw / h }
  -uniroot(f, c(-14, 0), tol = 1e-12)$root
}

# ---- 2. Three ways to the same answer --------------------------------
x_short <- sqrt(Ka * C)                          # x << C
x_quad  <- (-Ka + sqrt(Ka^2 + 4 * Ka * C)) / 2   # exact for the acid
pct     <- x_quad / C * 100
cat(sprintf("Shortcut : [H+] = %.3e M   pH = %.2f\n", x_short, ph(x_short)))
cat(sprintf("Quadratic: [H+] = %.3e M   pH = %.2f\n", x_quad, ph(x_quad)))
cat(sprintf("Exact    : pH = %.3f (water included)\n", ph_weak_acid_exact(C, Ka)))
cat(sprintf("Percent ionised = %.2f%%  ->  5%% rule %s\n", pct, ifelse(x_short / C < 0.05, "passed", "FAILED")))
cat(sprintf("pOH = %.2f\n", 14 - ph(x_short)))

# ---- 3. Checks ----------------------------------------------------------
stopifnot(round(ph(x_short), 2) == 2.87,
          round(ph(x_quad), 2) == 2.88,
          abs(x_quad^2 / (C - x_quad) - Ka) < 1e-12,       # Ka comes back out
          round(ph_strong_acid(0.010), 2) == 2.00,          # 0.010 M HCl
          round(14 + log10(0.01), 2) == 12.00,              # 0.01 M NaOH
          round(ph_strong_acid(1.0e-8), 2) == 6.98,         # very dilute HCl
          round(4.74 + log10(0.15 / 0.10), 2) == 4.92)      # acetate buffer
x <- 10^-2.87
stopifnot(round(x^2 / (0.10 - x) * 1e5, 1) == 1.8)          # Ka from pH 2.87
cat("Verdict: Verified (all checks pass)\n")

# ---- 4. Figure: pH of the weak acid and of HCl against concentration ----
concs <- 10^(-(0:24) / 4)                       # 1 M down to 1e-6 M
png("ph_curve.png", width = 2100, height = 1350, res = 300)
par(mar = c(4.5, 4.5, 3, 1))
plot(concs, sapply(concs, ph_weak_acid_exact, ka = Ka), log = "x", type = "l", lwd = 2.5,
     col = "#2563eb", xlim = rev(range(concs)), ylim = c(0, 7),
     xlab = "Concentration (mol/L)", ylab = "pH", main = "Weak and strong acid pH against concentration")
lines(concs, sapply(concs, ph_strong_acid), lwd = 2.5, col = "#dc2626")
points(C, ph(x_quad), pch = 19, cex = 1.5, col = "#ea580c")
legend("topleft", c("weak acid (Ka 1.8e-5)", "strong acid (HCl)", "your solution"),
       col = c("#2563eb", "#dc2626", "#ea580c"), lwd = c(2.5, 2.5, NA), pch = c(NA, NA, 19), bty = "n")
invisible(dev.off())
cat("Saved ph_curve.png\n")

Output: Shortcut : [H+] = 1.342e-03 M pH = 2.87, Quadratic: [H+] = 1.333e-03 M pH = 2.88, Exact : pH = 2.875, Percent ionised = 1.33%, Verdict: Verified.

10.2 R traps

• log() in R is the natural log. Use log10() for pH.
• uniroot() needs an interval where the function changes sign; searching log₁₀[H⁺] between −14 and 0 always works.
• From a pKa, write Ka <- 10^-4.74; 10^4.74 without the minus sign is a common slip that gives a huge Ka.

10.3 What each part does

10.4 Packages

For full speciation of mixtures and natural waters, the AquaEnv and seacarb packages compute pH from alkalinity and dissolved carbon, the same chemistry as the pool mode of this calculator.

10.5 The figure

The script saves ph_curve.png at 2100 × 1350 pixels and 300 dpi: pH against concentration for the weak acid and for HCl, with your solution marked.

This script does the same job in plain Python: shortcut, quadratic, exact charge balance by bisection, percent ionisation, the 5% rule, eight checks (including 0.010 M HCl, 0.01 M NaOH and a buffer) and a 300 dpi chart. It is the same method the pH calculator runs in your browser.

Install once: pip install matplotlib

11.1 The complete script

Python, saves ph_curve.png

# ============================================================
# pH calculator in Python: weak acid pH by the shortcut, the
# quadratic and the exact charge balance (water included)
# StatsUnlock - https://statsunlock.com/ph-calculator/
# pip install matplotlib
# ============================================================
import math
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

# ---- 1. Your solution -------------------------------------------------
C = 0.10          # mol/L of the weak acid
Ka = 1.8e-5       # acid dissociation constant (use Ka = 10**-pKa if you have pKa)
Kw = 1.0e-14      # ion product of water at 25 degrees C

def ph(h):
    return -math.log10(h)

def ph_strong_acid(c, n=1):
    """Strong acid, water included: [H+] = (nC + sqrt((nC)^2 + 4Kw)) / 2."""
    a = n * c
    return ph((a + math.sqrt(a * a + 4 * Kw)) / 2)

def ph_weak_acid_exact(c, ka):
    """Solve [H+] = Ka*C/(Ka+[H+]) + Kw/[H+] by bisection on log10[H+]."""
    lo, hi = -14.0, 0.0
    for _ in range(200):
        mid = (lo + hi) / 2
        h = 10 ** mid
        f = h - ka * c / (ka + h) - Kw / h        # charge balance
        lo, hi = (lo, mid) if f > 0 else (mid, hi)
    return -((lo + hi) / 2)

# ---- 2. Three ways to the same answer --------------------------------
x_short = math.sqrt(Ka * C)                                # x << C
x_quad = (-Ka + math.sqrt(Ka ** 2 + 4 * Ka * C)) / 2       # exact for the acid
pct = x_quad / C * 100
print(f"Shortcut : [H+] = {x_short:.3e} M   pH = {ph(x_short):.2f}")
print(f"Quadratic: [H+] = {x_quad:.3e} M   pH = {ph(x_quad):.2f}")
print(f"Exact    : pH = {ph_weak_acid_exact(C, Ka):.3f} (water included)")
print(f"Percent ionised = {pct:.2f}%  ->  5% rule {'passed' if x_short / C < 0.05 else 'FAILED'}")
print(f"pOH = {14 - ph(x_short):.2f}")

# ---- 3. Checks ----------------------------------------------------------
assert round(ph(x_short), 2) == 2.87
assert round(ph(x_quad), 2) == 2.88
assert abs(x_quad ** 2 / (C - x_quad) - Ka) < 1e-12           # Ka comes back out
assert round(ph_strong_acid(0.010), 2) == 2.00                # 0.010 M HCl
assert round(14 + math.log10(0.01), 2) == 12.00               # 0.01 M NaOH
assert round(ph_strong_acid(1.0e-8), 2) == 6.98               # very dilute HCl
assert round(4.74 + math.log10(0.15 / 0.10), 2) == 4.92       # acetate buffer
x = 10 ** -2.87
assert round(x * x / (0.10 - x) * 1e5, 1) == 1.8              # Ka from pH 2.87
print("Verdict: Verified (all checks pass)")

# ---- 4. Figure: pH of the weak acid and of HCl against concentration ----
concs = [10 ** (-k / 4) for k in range(0, 25)]                # 1 M down to 1e-6 M
fig, ax = plt.subplots(figsize=(7, 4.5))
ax.plot(concs, [ph_weak_acid_exact(c, Ka) for c in concs], color="#2563eb", lw=2.5, label="weak acid (Ka 1.8e-5)")
ax.plot(concs, [ph_strong_acid(c) for c in concs], color="#dc2626", lw=2.5, label="strong acid (HCl)")
ax.scatter([C], [ph(x_quad)], color="#ea580c", zorder=5, s=60, label=f"your solution, pH {ph(x_quad):.2f}")
ax.set_xscale("log"); ax.invert_xaxis()
ax.set_xlabel("Concentration (mol/L)"); ax.set_ylabel("pH")
ax.set_title("Weak and strong acid pH against concentration")
ax.legend(); fig.tight_layout()
fig.savefig("ph_curve.png", dpi=300)
print("Saved ph_curve.png")

Output: Shortcut : [H+] = 1.342e-03 M pH = 2.87, Quadratic: [H+] = 1.333e-03 M pH = 2.88, 5% rule passed, Verdict: Verified.

11.2 Python traps

• math.log() is the natural log; use math.log10().
• Write 1.8 × 10⁻⁵ as 1.8e-5; 1.8*10^-5 is a bitwise XOR in Python and gives nonsense.
• Bisect on log₁₀[H⁺], not on [H⁺] itself, so very small values converge.

11.3 Packages

pip install pHcalc calculates the pH of mixtures of acids, bases and salts from their pKa values, and chempy solves general equilibrium systems.

11.4 What each part does

11.5 The figure

The script saves ph_curve.png at 300 dpi: weak acid and strong acid pH against concentration on a log axis.

12.1 pH, pOH, [H⁺] and [OH⁻] at 25 °C

12.2 Ka, pKa, Kb and pKb of common weak acids and bases

Typical textbook values at 25 °C (values in different books vary in the second figure). The pH column is for a 0.10 M solution, computed by the same engine as the pH calculator.

12.3 pH of strong acids and bases at common concentrations

H₂SO₄ uses the textbook assumption of two H⁺. The last column includes water: a very dilute acid or base cannot cross pH 7.

12.4 pH and pOH calculations worksheet with answers

Twenty-five questions in three groups: pH and pOH from concentrations, weak acids and bases, and buffers and Ka. Every answer was checked by the calculator engine. Cover the last column and try them first.

Eight examples run through the calculator, from hydrochloric acid to a swimming pool. Each figure is drawn from the computed numbers: the marker shows the pH on a 0 to 14 scale.

This is the written method examiners expect, on the same problem as Sections 7, 9, 10 and 11: calculate the pH of 0.10 M acetic acid, Ka = 1.8 × 10⁻⁵. Simpler cases come first.

14.1 How to calculate pH from molarity (strong acid)

A strong acid is fully ionised, so [H⁺] = its molarity (times the number of H⁺). For 0.010 M HCl: pH = −log(0.010) = 2.00. For 0.050 M H₂SO₄ (textbook, two H⁺): [H⁺] = 0.10 M, pH = 1.00.

14.2 How to calculate the pH of a strong base

Find [OH⁻], then pOH, then pH. 0.01 M sodium hydroxide: pOH = −log(0.01) = 2.00, so pH = 14.00 − 2.00 = 12.00. For 0.001 M NaOH, pOH = 3.00 and pH = 11.00. Ba(OH)₂ gives two OH⁻: 0.0050 M has [OH⁻] = 0.010 M and pH 12.00.

14.3 How to calculate the pH of a weak acid, step by step

1. Write the equilibrium. CH₃COOH ⇌ H⁺ + CH₃COO⁻.
2. ICE table. Initial 0.10, 0, 0; change −x, +x, +x; equilibrium 0.10 − x, x, x.
3. Write Ka. 1.8 × 10⁻⁵ = x² ÷ (0.10 − x).
4. Approximate. If x ≪ 0.10, x² = 1.8 × 10⁻⁶ and x = 1.34 × 10⁻³ M.
5. Check the 5% rule. 1.34 × 10⁻³ ÷ 0.10 × 100 = 1.3%, under 5%: the approximation holds.
6. pH. pH = −log(1.34 × 10⁻³) = 2.87.
7. Check. The quadratic gives x = 1.33 × 10⁻³ M and pH 2.88; putting x back gives Ka = 1.8 × 10⁻⁵.

14.4 How to calculate Ka (or pKa) from pH

Turn the pH into x, then put x into the Ka expression. A 0.10 M acid has pH 2.87: x = 10^−2.87 = 1.35 × 10⁻³ M, so Ka = (1.35 × 10⁻³)² ÷ (0.10 − 0.00135) = 1.8 × 10⁻⁵ and pKa = 4.73. For a weak base, use pOH = 14 − pH to get [OH⁻] and find Kb the same way.

14.5 How to calculate the pH of a buffer

Use pH = pKa + log([A⁻] ÷ [HA]). 0.10 M acetic acid with 0.15 M sodium acetate (pKa 4.74): pH = 4.74 + log(1.5) = 4.74 + 0.18 = 4.92. Moles work as well as concentrations, because the volume cancels in the ratio.

14.6 How to calculate the pH at the equivalence point

For 25.00 mL of 0.100 M acetic acid titrated with 0.100 M NaOH, equivalence is at 25.00 mL and the flask then holds 0.00250 mol of acetate in 50.00 mL: 0.0500 M. Kb = Kw ÷ Ka = 10⁻¹⁴ ÷ 10^−4.76 = 5.8 × 10⁻¹⁰, [OH⁻] = √(5.8 × 10⁻¹⁰ × 0.0500) = 5.4 × 10⁻⁶ M, pOH = 5.27 and pH = 8.73.

14.7 On a scientific calculator

On a Casio fx-991EX or fx-82MS, pH from [H⁺]: (−) log 1.34 ×10^x (−) 3 = gives 2.872. [H⁺] from pH: SHIFT log (10^x) (−) 2.87 = gives 1.349 × 10⁻³. The shortcut: √ 1.8 ×10^x (−) 5 × 0.1 ) = gives 1.342 × 10⁻³.

Use log, not ln, and the negative key (−) for the exponent. Key names differ between models; check your manual.

14.8 Do it in the lab: find the Ka of acetic acid with a pH meter

Example data. The two methods agree within 0.02 pKa units. The half-equivalence method is usually better: a pH error of 0.01 changes the single-point Ka by about 5%, but the half-equivalence pKa only by 0.01.

14.9 How to calculate pH: practice problems with answers

Ten practice problems, easy to hard. Open each one to see the answer.

14.10 Mistakes that lose marks

Research on acid–base learning finds students mix up strength and concentration and treat pH as linear rather than logarithmic (Ross & Munby, 1991; Demircioğlu, Ayas & Demircioğlu, 2005).

• Using ln instead of log: 0.10 M HCl then comes out as pH 2.30.
• Reporting pOH as the pH of a base.
• Treating a weak acid as strong: pH 1.00 instead of 2.88 for 0.10 M acetic acid.
• Forgetting the 2 in H₂SO₄ or Ba(OH)₂.
• Skipping the 5% check when Ka is large or C is small.
• Putting acid over base in Henderson–Hasselbalch: it is log(base ÷ acid).
• Saying 10⁻⁸ M HCl has pH 8: an acid cannot make water basic.

Four real examples

Decision tree

1. Do you already have pH, pOH, [H⁺] or [OH⁻]? Yes: conversion mode. No: go on.
2. Is the acid or base strong (HCl, HNO₃, H₂SO₄, NaOH, KOH)? Yes: strong mode.
3. Is there only a weak acid or base? Yes: weak mode with Ka or Kb. Both the acid and its conjugate base? Buffer mode.
4. Is a titrant being added? Yes: titration mode. Know the pH and want Ka? Ka mode.

Limitations. pH is formally defined from the activity of H⁺ and measured against standard buffers (Buck et al., 2002; Covington, Bates & Durst, 1985). This calculator works with concentrations, which is what courses use and what is accurate in dilute solutions. Kw at other temperatures follows Bandura and Lvov (2006) closely through the Harned and Owen fit used here.

What the answer is

pH = −log[H⁺]. For a strong acid, [H⁺] is the molarity; for a weak acid, [H⁺] ≈ √(Ka C). 0.10 M acetic acid has pH 2.87 by the shortcut and 2.88 exactly.

What it means chemically

pH tells you the hydrogen ion concentration on a logarithmic scale. Each unit is a factor of ten, so small pH changes are large chemical changes.

Watch out for

Use log, not ln; report pH, not pOH, for bases; check the 5% rule; put base over acid in Henderson–Hasselbalch; and include water for very dilute solutions.

Next steps

Practise with the worksheet in Section 12.4 and the ten problems in Section 14.9, then use the buffer and titration modes to see how pH responds as acid or base is added.

Logarithms and quadratics are algebra, and comparing measured with calculated pH is statistics; these tools cover both.

Peer-reviewed sources for the methods and data used in this pH calculator: the IUPAC definition and measurement of pH, the ionisation of water, buffer equations and data, carbonate chemistry, and research on how students learn acids, bases and pH.

1. Bandura, A. V., & Lvov, S. N. (2006). The ionization constant of water over wide ranges of temperature and density. Journal of Physical and Chemical Reference Data, 35(1), 15–30. https://doi.org/10.1063/1.1928231
2. Buck, R. P., Rondinini, S., Covington, A. K., Baucke, F. G. K., Brett, C. M. A., Camoes, M. F., ... Wilson, G. S. (2002). Measurement of pH. Definition, standards, and procedures (IUPAC Recommendations 2002). Pure and Applied Chemistry, 74(11), 2169–2200. https://doi.org/10.1351/pac200274112169
3. Camacho, M., & Good, R. (1989). Problem solving and chemical equilibrium: Successful versus unsuccessful performance. Journal of Research in Science Teaching, 26(3), 251–272. https://doi.org/10.1002/tea.3660260306
4. Covington, A. K., Bates, R. G., & Durst, R. A. (1985). Definition of pH scales, standard reference values, measurement of pH and related terminology (Recommendations 1984). Pure and Applied Chemistry, 57(3), 531–542. https://doi.org/10.1351/pac198557030531
5. Demircioğlu, G., Ayas, A., & Demircioğlu, H. (2005). Conceptual change achieved through a new teaching program on acids and bases. Chemistry Education Research and Practice, 6(1), 36–51. https://doi.org/10.1039/B4RP90003K
6. Drechsler, M., & Schmidt, H.-J. (2005). Textbooks’ and teachers’ understanding of acid-base models used in chemistry teaching. Chemistry Education Research and Practice, 6(1), 19–35. https://doi.org/10.1039/B4RP90002B
7. Goldberg, R. N., Kishore, N., & Lennen, R. M. (2002). Thermodynamic quantities for the ionization reactions of buffers. Journal of Physical and Chemical Reference Data, 31(2), 231–370. https://doi.org/10.1063/1.1416902
8. Hawkes, S. J. (1992). Arrhenius confuses students. Journal of Chemical Education, 69(7), 542. https://doi.org/10.1021/ed069p542
9. Henderson, L. J. (1908). Concerning the relationship between the strength of acids and their capacity to preserve neutrality. American Journal of Physiology, 21(2), 173–179. https://doi.org/10.1152/ajplegacy.1908.21.2.173
10. Kolb, D. (1978). Acids and bases. Journal of Chemical Education, 55(7), 459. https://doi.org/10.1021/ed055p459
11. Millero, F. J. (1995). Thermodynamics of the carbon dioxide system in the oceans. Geochimica et Cosmochimica Acta, 59(4), 661–677. https://doi.org/10.1016/0016-7037(94)00354-O
12. Nakhleh, M. B. (1992). Why some students don’t learn chemistry: Chemical misconceptions. Journal of Chemical Education, 69(3), 191–196. https://doi.org/10.1021/ed069p191
13. Nakhleh, M. B., & Krajcik, J. S. (1994). Influence of levels of information as presented by different technologies on students’ understanding of acid, base, and pH concepts. Journal of Research in Science Teaching, 31(10), 1077–1096. https://doi.org/10.1002/tea.3660311004
14. Nurrenbern, S. C., & Pickering, M. (1987). Concept learning versus problem solving: Is there a difference? Journal of Chemical Education, 64(6), 508–510. https://doi.org/10.1021/ed064p508
15. Orgill, M., & Sutherland, A. (2008). Undergraduate chemistry students’ perceptions of and misconceptions about buffers and buffer problems. Chemistry Education Research and Practice, 9(2), 131–143. https://doi.org/10.1039/B806229N
16. Plummer, L. N., & Busenberg, E. (1982). The solubilities of calcite, aragonite and vaterite in CO₂-H₂O solutions between 0 and 90°C, and an evaluation of the aqueous model for the system CaCO₃-CO₂-H₂O. Geochimica et Cosmochimica Acta, 46(6), 1011–1040. https://doi.org/10.1016/0016-7037(82)90056-4
17. Po, H. N., & Senozan, N. M. (2001). The Henderson-Hasselbalch equation: Its history and limitations. Journal of Chemical Education, 78(11), 1499. https://doi.org/10.1021/ed078p1499
18. Ross, B., & Munby, H. (1991). Concept mapping and misconceptions: A study of high-school students’ understandings of acids and bases. International Journal of Science Education, 13(1), 11–23. https://doi.org/10.1080/0950069910130102
19. Sheppard, K. (2006). High school students’ understanding of titrations and related acid-base phenomena. Chemistry Education Research and Practice, 7(1), 32–45. https://doi.org/10.1039/B5RP90014J
20. Watters, D. J., & Watters, J. J. (2006). Student understanding of pH: “I don’t know what the log actually is, I only know where the button is on my calculator”. Biochemistry and Molecular Biology Education, 34(4), 278–284. https://doi.org/10.1002/bmb.2006.494034042628