Difficulty: Intermediate | Prerequisites: Beer's Law theory (see Part 1 study notes), logarithms, molarity, dilution

TL;DR

This set of notes walks through the worksheet calculations step by step: converting %T data to absorbance, plotting an absorption spectrum to find the best measurement wavelength, building a Beer's Law calibration curve from standard solutions, and using the curve to calculate the mass of an analyte in a real sample. These are the exact problem types that appear on the worksheet and in exams.

Key Terms

Absorption spectrum

A plot of absorbance (y-axis) vs. wavelength (x-axis) for a substance. The peak of the curve indicates the analytical wavelength. Think of it as a fingerprint showing which colours of light the substance soaks up most.

Optimum analytical wavelength

The wavelength at the absorbance maximum of the spectrum. Measuring here gives the greatest sensitivity to concentration changes.

Best-fit line (linear regression)

The straight line that minimises the squared distances between the data points and the line. Its equation y = mx + b gives slope (m) and y-intercept (b), which you use to solve for unknowns.

Mass from concentration

Once you know C (mol/L) from the Beer's Law plot, mass in grams = C × V (in litres) × molar mass (g/mol). This is the final step when the question asks for grams of analyte.

Core Content: Worked Problems

Worked Problem 1: Absorption Spectrum of the Cu⁺-Cuproine Complex

The worksheet gives %T values at 14 wavelengths for the Cu⁺-cuproine complex in 3-methyl-1-butanol. The task has three parts.

Part (a): Convert %T to absorbance and plot the spectrum

Apply A = 2 − log₁₀(%T) to each data point.

Sample calculation for λ = 360 nm: A = 2 − log₁₀(2.51) = 2 − 0.3997 = 1.600.

Plot A (y-axis) vs. λ (x-axis). The curve should show a large peak near 360 nm (UV edge) and a second, broader peak around 545 to 555 nm in the visible region.

Part (b): Suggest the optimum analytical wavelength

The visible absorption maximum sits at approximately 550 nm (A = 0.851). This is where absorbance is highest in the visible range, so 550 nm is the optimum analytical wavelength for Beer's Law measurements of this complex.

The 360 nm peak has a higher absorbance, but it sits at the UV edge of the visible range and may not be accessible on all visible-range spectrophotometers. The question asks about the visible spectrum, so 550 nm is the answer.

Part (c): Is distilled water an appropriate blank?

No. The Cu⁺-cuproine complex is dissolved in 3-methyl-1-butanol, not water. The blank must match the solvent and matrix of the sample. Therefore 3-methyl-1-butanol is the appropriate reference solution. Using water would introduce a mismatch in refractive index and solvent absorbance, giving incorrect readings.

Worked Problem 2: Beer's Law Plot and Unknown Mass Calculation

Five standard Cu⁺-cuproine solutions in 3-methyl-1-butanol were measured at the analytical wavelength (550 nm). The data are given as %T.

Step 1: Convert %T to absorbance

Step 2: Plot A vs. C and find the best-fit line

Plot absorbance (y-axis) vs. concentration (x-axis). The five points should fall close to a straight line. Using linear regression, the equation is approximately:

y = 6275x − 0.008 (your exact values will vary with rounding)

R² should be ≥ 0.98, confirming Beer's Law holds.

Step 3: Find the unknown concentration

The Cu⁺ ion from 100. mL of wine was complexed and extracted. The %T of this solution at 550 nm was 47.3%.

Convert: A = 2 − log₁₀(47.3) = 2 − 1.6749 = 0.325.

Using the best-fit equation: C = (A − b) / m = (0.325 − (−0.008)) / 6275 = 0.333 / 6275 ≈ 5.31 × 10⁻⁵ mol/L.

Step 4: Calculate the mass of Cu⁺ in the wine sample

Since 1 mol of Cu⁺-cuproine contains 1 mol Cu⁺, the Cu⁺ concentration is also 5.31 × 10⁻⁵ mol/L.

The volume of the solution is 100. mL = 0.100 L.

moles Cu⁺ = 5.31 × 10⁻⁵ mol/L × 0.100 L = 5.31 × 10⁻⁶ mol

mass Cu⁺ = 5.31 × 10⁻⁶ mol × 63.55 g/mol = 3.37 × 10⁻⁴ g

So approximately 3.4 × 10⁻⁴ g (0.34 mg) of Cu⁺ ion was present in 100. mL of the wine.

Note: your numerical answer will differ slightly depending on the exact regression coefficients your software gives. The method is what matters.

Formulas

A = 2 - \log_{10}(\%T)

Converts percent transmittance to absorbance. This is the first step in every worksheet problem that gives you %T data.

y = mx + b

The Beer's Law plot trendline, where y = absorbance, x = concentration (mol/L), m = slope (equal to εb), and b = y-intercept.

C_{\text{unknown}} = \frac{A_{\text{unknown}} - b}{m}

Rearranged trendline equation. Plug in the unknown's absorbance to get its concentration.

\text{mass (g)} = C \times V \times M

Final step: concentration (mol/L) × volume (L) × molar mass (g/mol) = mass in grams. For Cu⁺, M = 63.55 g/mol.

Common Misconceptions

• Students sometimes use natural log (ln) instead of log₁₀ when converting %T to absorbance. The formula requires log base 10. Using ln gives the wrong answer.

• When reading concentration from the Beer's Law plot, students sometimes forget to account for the y-intercept and just divide A by the slope. Always use the full equation: C = (A − b) / m.

• Students occasionally confuse the volume of the wine sample (100. mL) with the volume of the extracted solution. Read the problem carefully to identify which volume to use in the mass calculation.

• Some students plot %T directly against concentration and try to draw a straight line. The linear relationship is between absorbance and concentration, not %T and concentration. Convert first, then plot.

Why It Matters / Exam Flags

⚠️ The %T → A conversion is virtually guaranteed to appear. Practise until you can do A = 2 − log₁₀(%T) without looking it up.

⚠️ Identifying the analytical wavelength from a spectrum is a standard short-answer question. Look for the highest absorbance peak in the relevant region.

⚠️ "Is distilled water the appropriate blank?" is a favourite question. The answer depends on the solvent: match the blank to the sample matrix.

⚠️ The final mass calculation chains together several steps (convert %T → A, read C from the plot, then mass = C × V × M). Show every step clearly for full marks; skipping intermediate values is a common way to lose points.

Practice Q&A

Q: A solution has a %T of 22.0 at 540 nm. What is the absorbance?

A: A = 2 − log₁₀(22.0) = 2 − 1.342 = 0.658.

Q: Your Beer's Law plot gives the equation A = 5800C + 0.012. A solution of unknown concentration has an absorbance of 0.500. What is the concentration?

A: C = (0.500 − 0.012) / 5800 = 0.488 / 5800 = 8.41 × 10⁻⁵ mol/L.

Q: Using the concentration from the previous question, calculate the mass of Cu⁺ in 0.100 L of solution. (Molar mass of Cu = 63.55 g/mol.)

A: mass = 8.41 × 10⁻⁵ mol/L × 0.100 L × 63.55 g/mol = 5.35 × 10⁻⁴ g (about 0.54 mg).

Q: A student plots %T vs. concentration instead of absorbance vs. concentration and gets a curved line. Why is the plot not linear?

A: Beer's Law gives a linear relationship between absorbance and concentration, not between %T and concentration. Because A = −log₁₀(T), the relationship between T (or %T) and C is exponential, producing a curve.

Q: An absorption spectrum shows two peaks, one at 370 nm (A = 1.45) and one at 545 nm (A = 0.82). The spectrophotometer only works in the 400 to 700 nm range. Which wavelength should be used for quantitative analysis?

A: 545 nm. Although the 370 nm peak has higher absorbance, it falls outside the instrument's operating range. The 545 nm peak is the highest absorbance within the accessible range and therefore gives the best sensitivity.

Related Terms / Search Tags

Beer's Law worksheet, spectrophotometry calculations, %T to absorbance conversion, absorption spectrum, analytical wavelength, Cu-cuproine complex, 3-methyl-1-butanol, Beer-Lambert plot, calibration curve, linear regression, R-squared, unknown concentration, mass from molarity, molar mass of copper, KMnO₄ spectroscopy lab, Purdue general chemistry, visible spectroscopy, colorimetric analysis