Problem 90 - Master Resource Book in JEE Main Physics

Question

\(80 \mathrm{~g}\) of impure sugar, when dissolved in a litre of water gives an optical rotation of \(9.9^{\circ}\), when placed in a tube of length \(20 \mathrm{~cm}\). If, the specific rotation of sugar is \(66^{\circ}\), then concentration of sugar solution will be (a) \(80 \mathrm{gL}^{-1}\) (b) \(75 \mathrm{gL}^{-1}\) [c) \(65 \mathrm{gL}^{-1}\) (d) \(50 \mathrm{gL}^{-1}\)

Step-by-Step Solution

Answer

The concentration of the sugar solution is (b) 75 gL^{-1}.

Step 1: Understand the Problem

We need to find the concentration of the sugar solution. We are given the optical rotation, the length of the tube, and the specific rotation of sugar.

Step 2: Use the Formula for Optical Rotation

The formula for optical rotation is \( \text{Observed Rotation} = [\alpha] \times c \times l \), where \([\alpha]\) is the specific rotation, \(c\) is the concentration in \(\mathrm{gL}^{-1}\), and \(l\) is the length of the tube in decimeters.

Step 3: Convert Length from cm to dm

The length of the tube is given as 20 cm. Convert this to decimeters (dm) since the formula requires the length in dm: \( l = 20 \text{ cm} = 2 \text{ dm} \).

Step 4: Substitute Known Values into the Formula

Substitute the known values into the formula: \( 9.9 = 66 \times c \times 2 \).

Step 5: Solve for Concentration

Solve for \( c \) by dividing both sides of the equation by \( 66 \times 2 \): \( c = \frac{9.9}{66 \times 2} = \frac{9.9}{132} \).

Step 6: Calculate the Concentration

Calculate \( c \) to estimate its value: \( c \approx \frac{9.9}{132} \approx 0.075 \; \mathrm{gL}^{-1} \). Convert this to g/L by multiplying by 1000: \( c = 75 \; \mathrm{gL}^{-1} \).

Step 7: Match with Options

Compare the calculated concentration with the given options: (a) \(80 \; \mathrm{gL}^{-1}\), (b) \(75 \; \mathrm{gL}^{-1}\), (c) \(65 \; \mathrm{gL}^{-1}\), (d) \(50 \; \mathrm{gL}^{-1}\). The concentration \(75 \; \mathrm{gL}^{-1}\) matches option (b).

Key Concepts

Understanding Specific Rotation

Specific rotation is a key concept in understanding how substances like sugar interact with polarized light. It is a measure of how much a substance can rotate the plane of polarization of light that passes through it.
This measure is specific to each substance and is usually indicated by the symbol \([\alpha]\).
Specific rotation is expressed in degrees and reported at a specific temperature and wavelength of light.- **Formal Definition:** Specific rotation is defined as the angle of rotation caused by a 1 g/mL solution of an optically active compound when the light path is 1 decimeter (dm) long.- **Formula:** The formula is \([\alpha] = \frac{\alpha}{c \times l}\), where \([\alpha]\) is the specific rotation, \(\alpha\) is the observed rotation, \(c\) is the concentration of the substance in grams per milliliter (g/mL), and \(l\) is the path length in decimeters (dm).Specific rotation helps to identify substances and test their purity. In this context, knowing the specific rotation of sugar assists in accurately calculating its concentration when solving problems involving optical activity.

Determining Sugar Concentration

Sugar concentration in a solution is essential for both practical and analytical purposes, particularly in fields like food science and chemistry.
In our problem, finding the sugar concentration involves understanding how much sugar is present in a certain volume of solution.The concentration is often expressed in grams per liter \((\mathrm{gL}^{-1})\). To determine the sugar concentration in a solution where the specific rotation and the observed rotation are known:- **Use the Formula:** The relationship is \(\alpha = [\alpha] \times c \times l\). Solving for \(c\), we get \(c = \frac{\alpha}{[\alpha] \times l}\).- **Convert Units:** Ensure consistent units; the path length should be in decimeters, as required by the formula.- **Calculate:** Substitute the known values into the formula to find the concentration \(c\). This calculation provides the amount of sugar per liter, which can help verify its purity or composition.Establishing the concentration is pivotal for industries and laboratory settings to ensure the quality and consistency of solutions.

Exploring Polarimetry

Polarimetry is the scientific technique used to measure the angle of rotation caused by passing polarized light through an optically active substance.
This method is crucial in fields like chemistry and biochemistry to study chiral molecules such as sugars, proteins, and amino acids. Key points of polarimetry include: - **Operation Mechanism:** A polarimeter is the instrument used, which consists of a light source, polarizer, sample tube, and analyzer. Light passes through the polarizer to become polarized before interacting with the optically active sample. - **Measuring Rotation:** When polarized light traverses through the solution, its plane is rotated. The degree of rotation is measured by the analyzer. Applications of polarimetry extend to various industries: - **Quality Control:** Ensuring the purity and concentration of substances such as pharmaceuticals and sweeteners. - **Research:** Aiding in structural analysis of organic compounds and new materials. In summary, polarimetry is a valuable tool in many scientific and industrial applications, providing insights into the composition and characteristics of materials through their interaction with light.