Calculate the pH of a 0.5 M (C₂H₅)₂NH Solution
Use this premium weak-base calculator to find pOH, pH, hydroxide concentration, protonated amine concentration, and percent ionization for diethylamine in water.
Weak Base pH Calculator
Enter or confirm the concentration and Kb for diethylamine, then click Calculate pH.
Expert Guide: How to Calculate the pH of a 0.5 M C₂H₅ 2NH Solution
If you want to calculate the pH of a 0.5 M C2H5 2NH solution, the first step is to interpret the formula correctly. In this context, the intended weak base is diethylamine, usually written as (C₂H₅)₂NH. Diethylamine is an amine, and amines behave as weak Brønsted bases in water because the nitrogen atom can accept a proton. That means the solution does not dissociate completely like sodium hydroxide. Instead, it establishes an equilibrium with water, producing a limited amount of hydroxide ions. The pH therefore depends on the base dissociation constant, Kb, and on the initial concentration of the weak base.
For a 0.5 M diethylamine solution, the chemistry is more interesting than simply saying the pH is “high.” To calculate it properly, you need the equilibrium reaction, the Kb expression, and either an exact quadratic solution or a justified weak-base approximation. This page gives you both. It also explains the underlying chemical reasoning so you can confidently solve textbook problems, homework assignments, lab pre-lab questions, and exam practice items involving weak organic bases.
1. Identify the equilibrium reaction
Diethylamine reacts with water according to the equilibrium:
The base accepts a proton from water, producing the conjugate acid diethylammonium and hydroxide ion. Because hydroxide ions are formed, the solution is basic and its pH is above 7.00 at 25°C.
2. Write the Kb expression
The base dissociation constant for this reaction is:
A commonly used room-temperature value for diethylamine is approximately 9.6 × 10-4. Because this Kb is much smaller than 1, diethylamine is a weak base, but it is still stronger than many other common weak bases used in introductory chemistry.
3. Set up an ICE table
To solve the equilibrium, use an ICE table, where I stands for initial, C for change, and E for equilibrium.
| Species | Initial (M) | Change (M) | Equilibrium (M) |
|---|---|---|---|
| (C₂H₅)₂NH | 0.500 | -x | 0.500 – x |
| (C₂H₅)₂NH₂⁺ | 0 | +x | x |
| OH⁻ | 0 | +x | x |
Substitute the equilibrium concentrations into the Kb expression:
This is the key equation. Solving for x gives the hydroxide concentration at equilibrium. Once you know x, the rest is straightforward:
- [OH⁻] = x
- pOH = -log[OH⁻]
- pH = 14.00 – pOH at 25°C
4. Solve exactly with the quadratic formula
For the exact method, rearrange the equation:
where C is the initial concentration, 0.500 M. Using Kb = 9.6 × 10-4:
- a = 1
- b = 9.6 × 10-4
- c = -(9.6 × 10-4)(0.500)
The physically meaningful root is:
Substituting values gives x ≈ 0.02145 M. Therefore:
- [OH⁻] ≈ 0.02145 M
- pOH ≈ 1.668
- pH ≈ 12.332
So the pH of a 0.5 M diethylamine solution is about 12.33 when Kb is taken as 9.6 × 10-4 at 25°C.
5. Can you use the weak-base approximation?
Yes, and this problem is a good example of when the approximation is reasonably close. If x is much smaller than 0.500, then 0.500 – x can be approximated as 0.500. The equation becomes:
Then:
That gives:
- pOH ≈ 1.659
- pH ≈ 12.341
The approximation differs from the exact answer by only about 0.009 pH units, which is negligible in many classroom settings. Still, the exact solution is preferred whenever precision matters or when you are checking the validity of the approximation.
6. Percent ionization and why it matters
Percent ionization tells you how much of the weak base actually reacts with water:
Using the exact result:
Since this is below 5%, the small-x approximation is justified. This is a useful checkpoint because many chemistry courses teach the 5% rule as a quick test for whether simplification is acceptable.
7. Comparison table: exact vs approximation for 0.500 M diethylamine
| Method | [OH⁻] (M) | pOH | pH | Approximation Error |
|---|---|---|---|---|
| Exact quadratic | 0.02145 | 1.668 | 12.332 | Reference value |
| Weak-base approximation | 0.02191 | 1.659 | 12.341 | About +0.009 pH units |
8. How concentration affects pH
One of the most useful ideas in acid-base equilibrium is that increasing the concentration of a weak base generally increases the hydroxide concentration and therefore raises the pH. However, the relationship is not linear because pH depends on the logarithm of [OH⁻]. In addition, weak-base systems are controlled by equilibrium, so doubling the base concentration does not double the pH shift.
The table below shows approximate exact-calculation values for diethylamine using Kb = 9.6 × 10-4 at 25°C. These values help you see the trend clearly.
| Initial Diethylamine Concentration (M) | Approx. Exact [OH⁻] (M) | pOH | pH | Percent Ionization |
|---|---|---|---|---|
| 0.050 | 0.00646 | 2.190 | 11.810 | 12.9% |
| 0.100 | 0.00933 | 2.030 | 11.970 | 9.33% |
| 0.500 | 0.02145 | 1.668 | 12.332 | 4.29% |
| 1.000 | 0.03050 | 1.516 | 12.484 | 3.05% |
Notice that as concentration increases from 0.050 M to 1.000 M, the pH rises, but not by a huge amount. That is because pH compresses concentration changes into a logarithmic scale. Also notice that percent ionization decreases at higher concentration, a common behavior for weak acids and weak bases.
9. Common mistakes students make
- Treating diethylamine like a strong base. A 0.5 M strong base would give pOH = 0.301 and pH = 13.699, which is much too high for this weak base.
- Using Ka instead of Kb. Amines are bases, so Kb is the direct equilibrium constant unless you are given the pKa of the conjugate acid.
- Forgetting the 25°C assumption. The relationship pH + pOH = 14.00 is exact only near 25°C for standard introductory calculations.
- Ignoring significant figures. Since Kb is typically given to two or three significant figures, your final pH should not imply unjustified precision.
- Skipping the percent ionization check. If the ionization is not small, the shortcut method may become inaccurate.
10. Why diethylamine is more basic than ammonia
Diethylamine is generally a stronger weak base than ammonia because alkyl groups attached to nitrogen donate electron density toward the nitrogen atom. That makes the lone pair more available for protonation. In aqueous solution, this tends to increase basicity, though steric and solvation factors also play a role. For many practical calculations, this means diethylamine solutions at the same molarity often have a higher pH than comparable ammonia solutions.
11. Practical interpretation of a pH near 12.33
A pH around 12.33 indicates a strongly basic laboratory solution. Even though diethylamine is classified as a weak base chemically, a 0.5 M solution still generates enough hydroxide to be corrosive and to produce a pronounced alkaline response in indicators. In real lab work, that means eye protection, gloves, and good ventilation are appropriate. From a measurement standpoint, a pH meter would likely show a value in the low 12s if the solution is freshly prepared and temperature is controlled near room temperature.
12. Step-by-step summary you can reuse
- Write the weak-base equilibrium reaction with water.
- Use the given initial concentration as the starting concentration of the base.
- Write an ICE table.
- Substitute equilibrium concentrations into the Kb expression.
- Solve for x exactly or by approximation.
- Set [OH⁻] = x.
- Compute pOH = -log[OH⁻].
- At 25°C, compute pH = 14.00 – pOH.
- Check percent ionization to judge the approximation.
13. Final answer
Using Kb = 9.6 × 10-4 for diethylamine at 25°C, the pH of a 0.5 M (C₂H₅)₂NH solution is approximately 12.33. The corresponding pOH is about 1.67, the hydroxide concentration is about 2.15 × 10-2 M, and the percent ionization is about 4.29%.
This result is the chemically correct way to answer the question “calculate the pH of a 05 m c2h5 2nh solution.” If you use the interactive calculator above, you can also test how changing Kb or concentration affects the final pH and compare exact versus approximate methods instantly.
14. Further reading from authoritative sources
For additional background on acid-base equilibria, pH measurement, and weak-base chemistry, review these educational references: