Question

A solution containing \(6.15 \mathrm{~g}\) of cholesterol per \(100 \mathrm{ml}\) of solution was made. When a portion of this solution was placed in a 5 -cm polarimeter tube, the observed rotation is \(-1.2^{\circ} .\) Determine the specific rotation of cholesterol. nes

Short answer

The specific rotation of cholesterol is approximately \(-39.02^\circ\).

Step-by-step solution

1. Calculate the concentration of the solution

We know that there are \(6.15~g\) of cholesterol in \(100~ml\) of the solution. We need to find the concentration in grams per milliliter: Concentration (c) = \(\frac{6.15~g}{100~ml}\) c = \(0.0615 \frac{g}{ml}\)

2. Convert tube length to decimeters

The length of the polarimeter tube is given in centimeters (5 cm). We need to convert this to decimeters (dm): \(Length~(l) = \frac{5~cm}{10} = 0.5~dm\)

3. Calculate the specific rotation

Now, we have everything we need to calculate the specific rotation (\(\alpha\)) using the formula mentioned earlier: \[\alpha = \frac{\theta}{l \times c}\] \(\alpha = \frac{-1.2^{\circ}}{0.5~dm \times 0.0615 \frac{g}{ml}}\) \(\alpha = \frac{-1.2^{\circ}}{0.03075~\frac{g}{ml}~dm}\) \(\alpha \approx -39.02^{\circ}\) So, the specific rotation of cholesterol is approximately \(-39.02^\circ\).

Key concepts

Solution Concentration

Solution concentration plays a crucial role in various chemical and biological processes and is pivotal in determining the specific rotation of a substance using a polarimeter. When we talk about concentration, we are referring to the amount of a substance (solute) present in a given volume of solution. It's often expressed in units such as grams per milliliter (g/mL), moles per liter (M), or weight/volume percent. In the exercise at hand, knowing the cholesterol concentration, which is 0.0615 g/mL, was the first step towards calculating the specific rotation. This value indicates how much cholesterol is present in each milliliter of the solution.

Understanding concentration not only affects the accuracy of scientific measurements but also ensures the repeatability of experiments. Given that specific rotation is an intrinsic property of substances, it's imperative that the concentration is measured precisely for consistent and reliable results. Imagine you're baking a cake - if the recipe calls for a specific amount of sugar, adding too much or too little would alter the taste significantly. Similarly, incorrect concentration values can lead to significant errors in specific rotation calculations.

Polarimeter

A polarimeter is an analytical tool used to measure the angle of rotation caused by passing polarized light through an optically active substance. This device is integral to the study of chiral molecules such as sugars, amino acids, and steroids like cholesterol. The way a polarimeter works is by shining polarized light through a sample solution where it interacts with the molecules. If the molecules are chiral, they will rotate the plane of polarized light to a certain degree, which can be left or right, known as levorotatory or dextrorotatory, respectively.

In our exercise, the polarimeter was used to determine that a cholesterol solution in a 5-cm tube rotated the light by -1.2 degrees, indicating that the cholesterol is levorotatory. This observational data is fundamental to calculating the specific rotation, which is a characteristic property of cholesterol. The accuracy of a polarimeter can be influenced by factors such as temperature, wavelength of light used, and calibration. Therefore, ensuring the polarimeter is correctly set up and calibrated is essential for obtaining precise measurements.

Specific Rotation Calculation

The specific rotation of a substance is a standard measure of its optical activity, defined as the rotation, in degrees, of the plane of polarized light per unit length of the sample tube and per unit concentration of the solution. The formula used to calculate specific rotation is \[\alpha = \frac{\theta}{l \times c}\] where \(\alpha\) represents the specific rotation, \(\theta\) is the observed rotation in degrees, \(l\) is the path length through the sample in decimeters, and \(c\) is the concentration in grams per milliliter.

It's analogous to taking a standard measure, regardless of the test sample size - think of it like obtaining a 'per square foot' cost of a house regardless of its total area. For cholesterol in our exercise, we use the observed rotation of -1.2 degrees, a sample tube of 0.5 dm, and a concentration of 0.0615 g/mL to calculate its specific rotation as -39.02 degrees. This calculation allows us to compare the optical activity of cholesterol with other substances in a standardized manner, which is especially useful in the fields of chemistry and biochemistry where optical properties are linked to molecular structure and purity.