Answer:
5.86
Explanation:
The equation for the equilibrium is
(CH₃)₂NH₂⁺ + H₂O ⇌ H₃O⁺ + (CH₃)₂NH; pKₐ= 10.73
For ease of typing, let’s rewrite the equation as
HA + H₂O ⇌ H₃O⁺ + A⁻
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Set up an ICE table:
HA + H₂O ⇌ H₃O⁺ + A⁻
I: 0.10 0 0
C: -x +x +x
E: 0.10-x x x
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Calculate Kₐ
Kₐ = 10^(-pKₐ) =10^(-10.73) = 1.86 × 10⁻¹¹
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Solve the Kₐ expression
Kₐ = {[H₃O⁺][A⁻]}/[HA] = 1.86 × 10⁻¹¹ Substitute values
(x × x)/(0.10-x) = 1.86 × 10⁻¹¹ Combine like terms
x²/(0.10-x) Check that x ≪ 0.10
0.10/(1.86 × 10⁻¹¹) = 5.37 × 10⁹ > 400 x ≪0.10; Ignore x
x²/0.10 = 1.86 × 10⁻¹¹ Cross-multiply
x² = 0.10 × 1.86 × 10⁻¹¹ Complete the multiplication
x² = 1.86 × 10⁻¹² Take the square root of both sides
x = [H₃O⁺] = 1.36 × 10⁻⁶ Take the negative log of each side
-log[H₃O⁺] = pH = -log(1.36 × 10⁻⁶) Complete the operation
pH = 5.86