$\varepsilon_{220}$ ($M^{-1}cm^{-1}$) $\varepsilon_{280}$ ($M^{-1}cm^{-1}$) Compound X $1000$ $200$ Compound Y $800$ $400$
Quantitative Analysis of Aqueous Solutions Using Spectrophotometry
This problem requires determining the concentration of compound Y in a mixture using absorbance data at two different wavelengths and known molar absorption coefficients ($\varepsilon$). The Beer-Lambert Law is the fundamental principle applied here, which states that the absorbance ($A$) of a solution is directly proportional to the concentration ($C$) of the absorbing species and the path length ($b$) of the light through the solution: $A = \varepsilon C b$
For a solution containing multiple absorbing species, the total absorbance at a given wavelength is the sum of the absorbances of each individual species. Since the path length ($b$) is given as $1$ cm, the Beer-Lambert Law simplifies to $A = \varepsilon C$.
Given Data
| Compound | $\varepsilon_{220}$ ($M^{-1}cm^{-1}$) | $\varepsilon_{280}$ ($M^{-1}cm^{-1}$) |
|---|---|---|
| X | $1000$ | $200$ |
| Y | $800$ | $400$ |
Absorbance Equations
Let $C_X$ be the concentration of compound X and $C_Y$ be the concentration of compound Y, both in Molarity (M).
At $220$ nm, the total absorbance ($A_{220}$) is given as $1.0$. Using the Beer-Lambert Law for the mixture:
$A_{220} = \varepsilon_{X,220} C_X + \varepsilon_{Y,220} C_Y$
Substituting the known values:
$1.0 = (1000 \, M^{-1}cm^{-1}) C_X + (800 \, M^{-1}cm^{-1}) C_Y \quad \quad (1)$
At $280$ nm, the total absorbance ($A_{280}$) is given as $0.4$. Similarly:
$A_{280} = \varepsilon_{X,280} C_X + \varepsilon_{Y,280} C_Y$
Substituting the known values:
$0.4 = (200 \, M^{-1}cm^{-1}) C_X + (400 \, M^{-1}cm^{-1}) C_Y \quad \quad (2)$
Solving for Concentration of Y
We have a system of two linear equations with two unknowns ($C_X$ and $C_Y$). We can solve this system:
- Isolate $C_X$ from Equation (2): $200 C_X = 0.4 - 400 C_Y$ $C_X = \frac{0.4 - 400 C_Y}{200}$ $C_X = 0.002 - 2 C_Y \quad \quad (3)$
- Substitute Equation (3) into Equation (1): $1.0 = 1000 (0.002 - 2 C_Y) + 800 C_Y$
- Simplify and solve for $C_Y$: $1.0 = 2 - 2000 C_Y + 800 C_Y$ $1.0 = 2 - 1200 C_Y$ $1200 C_Y = 2 - 1.0$ $1200 C_Y = 1.0$ $C_Y = \frac{1.0}{1200} \, M$
Unit Conversion
The question asks for the concentration of Y in millimolarity (mM). To convert from Molarity (M) to millimolarity (mM), multiply by $1000$.
$C_Y (\text{mM}) = C_Y (\text{M}) \times 1000$
$C_Y (\text{mM}) = \frac{1}{1200} \times 1000$
$C_Y (\text{mM}) = \frac{1000}{1200} = \frac{10}{12} = \frac{5}{6} \, \text{mM}$
Calculating the decimal value:
$C_Y (\text{mM}) \approx 0.8333 \, \text{mM}$
This calculated value falls within the range of $0.82$ to $0.84$ mM.
Important Questions from Enzyme Assays Molar Extinction Coefficient
- A solution shows a transmittance of 20% when taken in a cuvette of 2.5 cm path length. If the molar absorption coefficient of the solution is $12000 \text{ dm}^3/\text{mol.cm}$, the concentration of the solution is ________ $\times 10^5 \text{ mol/dm}^3$ (rounded off to two decimal places).
- A solution containing GTP has molar extinction coefficient of $1.55 \times 10^4$ $mol^{-1}dm^3cm^{-1}$ at a given wavelength. The concentration of GTP solution is $1.290 \times 10^{-5}$ $mol$ $dm^{-3}$. The absorbance of GTP solution in 1 cm cuvette at the same wavelength will be .................
- An enzyme preparation has activity of 2 Units per 20 $\mu$l, and protein concentration 0.4 mg/ml. The specific activity (Units/mg) of this enzyme will be ________
- Measurement of the absorbance of a solution containing NADH in a path length of 1cm cuvette at 340 nm shows the value of 0.31. The molar extinction coefficient of NADH is $6200 M^{-1} cm^{-1}$. The concentration of NADH in the solution is ________ $\mu M$ (correct to integer number).
- If a $10$ mM solution of a biomolecule in a cuvette of path length $10$ mm absorbs $90\%$ of the incident light at $280$ nm, the molar extinction coefficient of the biomolecule at this wavelength is ________ $M^{-1}cm^{-1}$. (Round off to two decimal places)
