Question

The weight of proteins or nucleic acids in solution is commonly determined by UV spectroscopy using the Beer-Lambert law. For example, the \(\varepsilon\) of double-stranded DNA at \(260 \mathrm{~nm}\) is \(6670 \mathrm{M}^{-1} \mathrm{~cm}^{-1}\). The formula weight of the repeating unit in DNA (650 Daltons on average) can be used as the molecular weight. What is the weight of DNA in \(2.0 \mathrm{~mL}\) of aqueous buffer if the absorbance, measured in a 1 -cm cuvette, is \(0.75\) ?

Short answer

Answer: The weight of DNA in the 2.0 mL aqueous buffer is approximately \(1.456 \times 10^{-4} \mathrm{~g}\).

Step-by-step solution

Understand the Beer-Lambert Law

The Beer-Lambert law is represented by the formula: \(A = \varepsilon \times c \times l\), where \(A\) is the absorbance, \(\varepsilon\) is the molar absorptivity constant (in \(\mathrm{M}^{-1} \mathrm{~cm}^{-1}\)), \(c\) is the concentration of the substance (in Molarity, M) and \(l\) is the path length (in cm). In this exercise, we are given the absorbance, molar absorptivity constant, and path length, and we need to find the concentration of the DNA solution.

Calculate the concentration of the DNA solution

Using the given values and the Beer-Lambert law formula, we can determine the concentration of the solution. Rearranging the formula to solve for the concentration, we get \(c = \frac{A}{\varepsilon \times l}\). Now, plug in the given values: \(A = 0.75\), \(\varepsilon = 6670 \mathrm{M}^{-1} \mathrm{~cm}^{-1}\), and \(l = 1 \mathrm{~cm}\). Calculate the concentration: \(c = \frac{0.75}{6670 \mathrm{M}^{-1} \mathrm{~cm}^{-1} \times 1 \mathrm{~cm}} = 1.12 \times 10^{-4} \mathrm{M}\).

Use the concentration and volume to find moles of DNA

Now that we have the concentration, we can use the volume of the solution (2.0 mL) to find the moles of DNA in the solution. Convert the volume from mL to L: \(2.0 \mathrm{~mL} = 0.002 \mathrm{~L}\). Then, use the concentration and volume to calculate moles of DNA: \(moles = concentration \times volume = 1.12 \times 10^{-4} \mathrm{M} \times 0.002 \mathrm{~L} = 2.24 \times 10^{-7} \mathrm{~moles}\).

Calculate the weight of DNA

Finally, to find the weight of DNA in the solution, we need to use the molecular weight of the repeating unit in DNA (650 Daltons). Convert the molecular weight to grams per mole (as 1 Dalton = 1 g/mol): \(650 \mathrm{~Daltons} = 650 \mathrm{~g/mol}\). Multiply the moles of DNA by the molecular weight: \(weight = moles \times molecular\ weight = 2.24 \times 10^{-7} \mathrm{~moles} \times 650 \mathrm{~g/mol} = 1.456 \times 10^{-4} \mathrm{~g}\). The weight of DNA in the 2.0 mL aqueous buffer is approximately \(1.456 \times 10^{-4} \mathrm{~g}\).