CHEM 370 | Week 4

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<h1 style = "text-align: left; font-weight: bold; margin-left: 175px;">Week 4: Standardization</h1>
<h2 style = "text-align: left; font-weight: bold; margin-left: 175px;">Determining $k_A$</h2>
<h5 style = "text-align: left; font-weight: bold; margin-left: 175px;">Harvey Chs 3, 4</h5>

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# Standardization

> **Standardization:** The process of determining the relationship between the amount of signal generated by a sample and the amount of analyte in the sample (determining $k\_A$)

$$
S\_A = k\_A C\_A
$$

> **Calibration:** The process of *adjusting* the signal to match a known value (changing $k\_A$).

???

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# Single vs. Multi-point Standardization

.image-credit[David Harvey / [Analytical Chemistry 2.1](https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Book%3A_Analytical_Chemistry_2.1_%28Harvey%29) / [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/3.0/at/deed.en)]

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# Multi-point Standardization

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# Linear Regression

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# Linear Regression

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# Practice

Determine $k\_A$ and $b_0$ for the following standards.  Calculate the concentration of Hg and 95% CI in a sample yielding three replicate signals of $x = [0.173, 0.166, 0.189]$.

| Concentration (μg / ml) |  Sₐ   |
|:-----------------------:|:-----:|
|          0.50           | 0.026 |
|          1.00           | 0.054 |
|          2.50           | 0.153 |
|          3.00           | 0.181 |
|          3.50           | 0.210 |

$$
s\_{C\_A} = \frac {s\_r} {k\_a} \sqrt{\frac {1} {m} + \frac {1} {n} + \frac {\left( \overline{S}\_{samp} - \overline{S}\_{std} \right)^2} {(k\_a)^2 \sum\_{i = 1}^{n} \left( C\_{std\_i} - \overline{C}\_{std} \right)^2}}
$$

???

kₐ = 0.06225
b₀ = -0.005933
R² = 0.99920

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# Standard Addition

In **standard addition** a standard is added to each sample to compensate for **matrix effects**.

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# Standard Addition: Single

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# Standard Addition: Multiple

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# Standard Addition: Multiple

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# Standard Addition: CI

The CI for concentration determined by standard addition is similar to external standards:

$$
s\_{C\_a} = \frac{s\_r}{k\_A} \sqrt{ \frac{1}{n} + \frac{(\overline{S}\_{STD})^2}{(k\_A)^2 \sum\_{i = 1}^n \left( C\_{STD, i} - \overline{C}\_{STD} \right)^2} }
$$

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# Internal Standards

What if, for example, some solvent evaporates from one standard but not the others?  Or from the sample?

> **Internal Standard:** A compound that is similar to -- but NOT identical to -- the analyte that is added in the same concentration to each sample and standard and used to compensate for changes in concentration during sample introduction, etc.

$$ S\_A = k\_A C\_A $$

$$ S\_{IS} = k\_{IS} C\_{IS} $$

$$\frac{ S\_A }{ S\_{IS} } = \frac{ k\_A C\_A }{ k\_{IS} C\_{IS} } = K \times \frac{ C\_A }{ C_{IS} }$$

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# Internal Standards

1. Add same concentration to each sample.
2. Plot $\frac{ S\_A }{ S\_{IS} }$ vs $C\_A$
3. Slope = $\frac{K}{C_{IS}}$

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# Internal Standard Example

A spectrophotometric method for the quantitative analysis of Pb$^{2+}$ in blood gives a linear internal standards calibration curve for which

$$\frac{ S\_A }{ S\_{IS} } = (2.11 \text{ppb}^{-1}) \times C\_A - 0.006$$

What is the concentration of Pb$^{2+}$ in a sample of blood if the analyte yields a signal of 3.36 and the internal standard gives a signal of 1.20?

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# Quality Control Samples

A **QC** or **Calibration Verification** standard is a sample made at a known concentration used to check the *accuracy* of the standard curve.

- Make near concentration of medium standard
- Use a different lot number, manufacturer, and/or stock solution
- Measured concentration should be ±10% of expected value (EPA Protocols)
- If not: check standard calculations, sources of gross error, and (last) contamination or manufacturing defects that occured with that specific lot.

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# Example Run List

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# Example Run List

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# Limit of Detection

???

# Limits of Detection

- Lower Limit of Detection

$$ LOD = S\_{mb} + 3s\_{mb} $$ or $$ LOD = \frac{3s\_{mb}}{k\_A} $$

- Lower Limit of Quantitation

$$ LOD = S\_{mb} + 10s\_{mb} $$ or $$ LOD = \frac{10s\_{mb}}{k\_A} $$

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# Upper Limit of Detection

The following data were gathered for sodium standards.  A blank run in triplicate returns $x = [-0.011, 0.005, 0.017]

| Concentration (μg / ml) |  Sₐ   |
|:-----------------------:|:-----:|
|          0.50           | 0.026 |
|          1.00           | 0.045 |
|          2.50           | 0.090 |
|          5.00           | 0.185 |
|          7.50           | 0.274 |
|          10.0           | 0.363 |
|          12.5           | 0.415 |
|          15.0           | 0.458 |
|          17.5           | 0.475 |

What is the LLOQ, ULOQ, and LDR?

???

Estimate LLOQ from $b_0$