Problem 74 - Analytical Chemistry

Question

The Stig Johansson pH calculator has been shown to give pH calculations of NIST standard buffers that are within a few thousandths of a \(\mathrm{pH}\) unit of the NIST values. The NIST buffers are given in Table 13.2 in Chapter \(13 .\) Use the calculator in Reference 15 to calculate the ActpH of the NIST phosphate buffer consisting of \(0.025 \mathrm{M} \mathrm{KH}_{2} \mathrm{PO}_{4}\) and \(0.025 \mathrm{M} \mathrm{Na}_{2} \mathrm{HPO}_{4}\) (footnote \(e\) ) at \(50^{\circ} \mathrm{C}\), and compare with the NIST value of \(6.833 .\) Use \(\mathrm{pK}_{w}=13.26, \mathrm{pK}_{1}=2.25\), \(\mathrm{p} K_{2}=7.18,\) and \(\mathrm{p} K_{3}=12.36\) for \(50^{\circ} \mathrm{C} .\) Don't forget to enter the temperature.

Step-by-Step Solution

Answer

The calculated pH is 7.18, differing from the NIST value 6.833. Use accurate computational tools for precision.

Step 1: Understanding the Problem

First, we need to calculate the pH of a phosphate buffer using the Stig Johansson pH calculator for a specific temperature condition.

Step 2: Identify the Buffer Components

The buffer is made up of two components: \(0.025\, M\, \mathrm{KH}_2\mathrm{PO}_4\) and \(0.025\, M\, \mathrm{Na}_2\mathrm{HPO}_4\), which correspond to the acid (\(\mathrm{H}_2\mathrm{PO}_4^-\)) and conjugate base (\(\mathrm{HPO}_4^{2-}\)) of the buffer system.

Step 3: Use the Henderson-Hasselbalch Equation

The pH of a buffer system is calculated using the Henderson-Hasselbalch equation: \[\mathrm{pH} = \mathrm{pK}_a + \log\left(\frac{[\mathrm{Conjugate\ Base}]}{[\mathrm{Acid}]}\right)\]For our system at \(50^\circ C\), the relevant \(\mathrm{pK}_a\) value is \(\mathrm{pK}_2 = 7.18\).

Step 4: Substitute Values into the Equation

Substitute \(\mathrm{pK}_2 = 7.18\), \([\mathrm{Conjugate\ Base}] = 0.025\, M\), and \([\mathrm{Acid}] = 0.025\, M\) into the equation: \[ \mathrm{pH} = 7.18 + \log\left(\frac{0.025}{0.025}\right) \]This simplifies to: \[ \mathrm{pH} = 7.18 + \log(1) = 7.18 \]

Step 5: Compare with the NIST Value

The calculated pH value at \(50^\circ C\) is \(7.18\). However, we need to compare this result with the NIST value of \(6.833\). It appears that the calculator's result will need adjustment considering the temperature-specific changes in the system, which is exactly what the Stig Johansson calculator is capable of handling.

Step 6: Summary and Conclusion

By comparing \(7.18\) with \(6.833\), the discrepancy shows that temperature adjustments are crucial for accurate pH. The Stig Johansson calculator will refine this calculation effectively to account for real-world deviations.

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a foundational concept in chemistry, especially when dealing with buffer solutions like phosphate buffers. This equation provides a simple way to calculate the pH of a buffer system. It is given by the formula:\[\text{pH} = \text{pK}_a + \log\left(\frac{[\text{Conjugate Base}]}{[\text{Acid}]}\right)\]
Here's how it works: The equation combines the acid dissociation constant (\text{pK}_a) of the weak acid in the buffer with the concentrations of the weak acid and its conjugate base.

• \(\text{pK}_a\) is a constant that indicates the strength of the acid; for a phosphate buffer at certain conditions, it's the value like \(\text{pK}_2 = 7.18\).
• The ratio \(\frac{[\text{Conjugate Base}]}{[\text{Acid}]}\) allows us to see the balance between the acid and its conjugate base.

The formula shows that pH is dependent on both \(\text{pK}_a\) and the ratio of concentrations, enabling accurate pH predictions for buffer solutions if the system's components are known.

Phosphate Buffer

Phosphate buffers are widely used because they maintain stable pH values over a range of conditions. They consist of mixtures of dihydrogen phosphate ions (\(\text{H}_2\text{PO}_4^-\)) which act as the acid, and hydrogen phosphate ions (\(\text{HPO}_4^{2-}\)) which serve as the conjugate base.
In the exercise's solution, the components:

• \(0.025\, M\, \text{KH}_2\text{PO}_4\) represents the acid part.
• \(0.025\, M\, \text{Na}_2\text{HPO}_4\) represents the conjugate base.

When creating a buffer, it's essential that these components are present in the right ratio, which is why in our example both concentrations are equal.
Phosphate buffers are particularly useful in biological applications because:

• They resist changes in pH upon dilution or addition of other components, up to a certain point.
• They provide a consistent environment for biochemical interactions at a specific pH, which is often around physiological pH.
• Temperature variations can still affect the pH, however buffer systems help mitigate drastic changes, maintaining stability.

Temperature Effects on pH

Temperature is a crucial factor to consider when calculating pH as it affects the dissociation of acids and bases. In this specific exercise, the temperature was set at \(50^\circ\text{C}\). Temperature can influence several aspects of a buffer solution:

• The ionization of water changes with temperature, reflected in the different values of \(\text{pK}_w\).
• The acid dissociation constants or \(\text{pK}_a\) values change with temperature, impacting the Henderson-Hasselbalch equation surrounding certain buffered systems like the phosphate buffer.

For instance, relocating our understanding of the \(\text{pK}_2\) value to \(7.18\) at \(50^\circ\text{C}\)—instead of its standard room temperature value—demonstrates how the equation's outcome can adjust with varied conditions. In practice, this means the calculated pH could divert from known standards like the NIST value of \(6.833\), highlighting this adaptation as necessary.
Advanced pH calculators, such as the Stig Johansson calculator, allow these temperature adjustments and offer more precise predictions by factoring in temperature's effects on all related constants.