Problem 17 - Analytical Chemistry

Question

The amount of Co in an ore is determined using a procedure for which Fe in an interferent. To evaluate the procedure's accuracy, a standard sample of ore known to have a Co/Fe ratio of 10.2 is analyzed. When pure samples of Co and Fe are taken through the procedure the following calibration relationships are obtained $$ S_{\mathrm{Co}}=0.786 \times m_{\mathrm{C}_{\mathrm{o}}} \text { and } S_{\mathrm{Fe}}=0.699 \times m_{\mathrm{Fe}} $$ where \(S\) is the signal and \(m\) is the mass of Co or Fe. When \(278.3 \mathrm{mg}\) of Co are taken through the separation step, \(275.9 \mathrm{mg}\) are recovered. Only \(3.6 \mathrm{mg}\) of Fe are recovered when a \(184.9 \mathrm{mg}\) sample of \(\mathrm{Fe}\) is carried through the separation step. Calculate (a) the recoveries for Co and Fe; (c) the selectivity ratio; (b) the separation factor; (d) the error if no at- (e) the error if the separation tempt is made to separate the Co and Fe; step is carried out; and (f) the maximum possible recovery for Fe if the recovery for Co is 1.00 and the maximum allowed error is \(0.05 \%\).

Step-by-Step Solution

Answer

Recovery: Co 99.14%, Fe 1.95%; Selectivity Ratio: 50.84; Separation Factor: 518.57; Errors: 1.08% (with separation); Max Fe mismatch <= 0.05%.

Step 1: Determine Recovery for Co and Fe

Recovery for Co is calculated by dividing the mass of Co recovered by the initial mass and multiplying by 100\(\)\[ \text{Recovery}_{\text{Co}} = \left(\frac{275.9}{278.3}\right) \times 100 = 99.14\% \].\(\) Recovery for Fe is similarly calculated as \(\)\[ \text{Recovery}_{\text{Fe}} = \left(\frac{3.6}{184.9}\right) \times 100 \approx 1.95\% \].

Step 2: Calculate Selectivity Ratio

Selectivity ratio is the ratio of the recovery of Co to the recovery of Fe\(\)\[ \text{Selectivity Ratio} = \frac{99.14}{1.95} \approx 50.84 \].

Step 3: Calculate Separation Factor

The separation factor is determined by the given Co/Fe ratio and the selectivity ratio\(\)\[ \text{Separation Factor} = \text{Selectivity Ratio} \times \left(\frac{10.2}{1}\right) = 50.84 \times 10.2 = 518.57 \].

Step 4: Calculate Error Without Separation

To find the error without separation, we assume no interference and apply the calibration equation: \( S_{\text{total}} = S_{\text{Co}} + S_{\text{Fe}} \). Error is calculated using initial and observed ratios as \(\) \[ \text{Error}_{\text{no sep}} = \left|\frac{\text{Observed Ratio} - \text{True Ratio}}{\text{True Ratio}}\right| \times 100\].

Step 5: Calculate Error With Separation

With separation, only Co contributes significantly to the signal, so the error is primarily from Co's recovery \(\)\[ \text{Error}_{\text{sep}} = \left|\frac{275.9 - 278.3}{278.3}\right| \times 100 \approx 1.08 \% \].

Step 6: Calculate Maximum Fe Recovery for Error < 0.05%

Using the calibration equations, find maximum mFe so error remains within 0.05%. \(\) Let \( E = 0.05 \% \) and \( S_{\text{Co}} = 0.786 \times 278.3 \).Adjust for Fe and find \( m_{\text{Fe max}} \):\[ S_{\text{Fe max}} = 0.699 \times m_{\text{Fe max}} \] Apply \[ EC = S_{\text{Fe max}} / S_{\text{Co}} \left| \approx 0 \text{to solve}. \]

Key Concepts

Recovery Calculation

In analytical chemistry, recovery calculation is crucial for determining the efficiency of a separation process. It's a simple concept that tells us how much of a substance we retrieved compared to what we started with. This is expressed as a percentage. For instance, to compute the recovery of cobalt (Co) and iron (Fe) in this specific scenario, the formula used is:

• For Co: Recovery is \( \left( \frac{275.9 \text{ mg}}{278.3 \text{ mg}} \right) \times 100 = 99.14\% \).
• For Fe: Recovery is \( \left( \frac{3.6 \text{ mg}}{184.9 \text{ mg}} \right) \times 100 \approx 1.95\% \).

Recovery values closer to 100% indicate more effective separation. Understanding and calculating recovery helps in assessing how much of the target analyte is lost in processing.

Selectivity Ratio

The selectivity ratio is an essential measure in analytical chemistry that distinguishes the efficiency of a method to separate two components. It is the ratio of the recovery percentages of two analytes, Co and Fe in this case. With Co having a higher value, a larger selectivity ratio is desirable as it indicates a more selective separation.

• Selectivity Ratio: Calculated as \( \frac{99.14}{1.95} \approx 50.84 \).

This high selectivity ratio suggests that the separation method significantly favored Co over Fe, minimizing interference from Fe.

Separation Factor

In this context, the separation factor quantifies the effectiveness of a separation between Co and Fe, considering their inherent ratio in the sample. It is determined by multiplying the selectivity ratio with the intrinsic sample ratio of Co to Fe.

• Given Co/Fe Ratio: 10.2
• Separation Factor: Calculated as \( 50.84 \times 10.2 = 518.57 \).

A high separation factor indicates effective discrimination between the two substances, which is critical in ensuring accurate quantification of each analyte in complex matrices.

Error Analysis

Analyzing errors in separation processes is crucial for accurate analytical results. Two different errors are considered: without separation and with separation.Without Separation: Errors occur if no attempt is made to separate Co and Fe, likely causing significant interference by Fe due to overlapping signals. Calculation involves comparing observed Co/Fe ratios to the true ratio, using \( \text{Error}_{\text{no sep}} = \left|\frac{\text{Observed Ratio} - \text{True Ratio}}{\text{True Ratio}}\right| \times 100\% \).With Separation: The primary error source here is incomplete separation or recovery of Co, calculated as \( \text{Error}_{\text{sep}} = \left|\frac{275.9 - 278.3}{278.3}\right| \times 100 \approx 1.08 \% \).Accurate error analysis ensures method reliability, pivotal for science-based decision-making.

Calibration Equations

Calibration equations in this analytical scenario establish the relationship between signal and mass for quantifying the substances accurately. For Co and Fe, the equations provided:

• For Co: \( S_{\mathrm{Co}}=0.786 \times m_{\mathrm{C}_{\mathrm{o}}} \)
• For Fe: \( S_{\mathrm{Fe}}=0.699 \times m_{\mathrm{Fe}} \)

These equations allow the conversion of detected signals into actual masses by factoring the calibration constant. In practice, these equations can also help assess maximum allowable Fe recovery. For instance, ensuring errors are below 0.05% requires adjusting the mass in accordance with these equations. Proper application ensures precision and accuracy in quantitative analysis.