Question

\(\mathrm{P}, \mathrm{Q}\) and \(\mathrm{R}\) are three optically active isomers in which Pand \(\mathrm{Q}\) are enantiomers. A solution of these isomers \(w^{2}\) prepared in which \(\mathrm{R}\) was \(50 \%\) and \(\mathrm{P}\) was \(40 \%\). Specific rotation of solution was measured to be \(-5^{\circ}\). Anothr solution of the same isomers but with different composition was also prepared in which \(\mathrm{R}\) was \(40 \%\) and \(\mathrm{Q}^{w}\) \(35 \%\). Specific rotation of this solution was found to be \(-8.5^{\circ}\). Assuming that measurements were perfornd with same polarimeter and at same temperature, the correct statement is (A) \(\mathrm{P}\) is levorotatory with \(-13^{\circ}\) specific angle of rotation (B) \(\mathrm{R}\) is levorotatory with \(-18^{\circ}\) specific angle of rotation. (C) \(Q\) is levorotatory with \(-13^{\circ}\) specific angle of rotation. (D) Magnitude of specific angle of rotation for \(\mathrm{P}\) and \(\mathrm{Q}\) is \(18^{\circ} .\)

Short answer

The correct statement is (D) Magnitude of specific angle of rotation for P and Q is 18°, since we determined that \(\alpha_P = 13°\), \(\alpha_Q = -7°\), and \(\alpha_R = -23°\).

Step-by-step solution

Solve for specific rotations in the first solution

Given that the first solution has 50% R, 40% P, and 10% Q (since total should be 100%), and its specific rotation is -5°. We can write: $$\alpha_{1} = \alpha_{P}(0.4) + \alpha_{Q}(0.1) + \alpha_{R}(0.5)$$ $$-5 = 0.4\alpha_{P} + 0.1\alpha_{Q} + 0.5\alpha_{R}$$ Where \(\alpha_{1}\) is the specific rotation of the first solution, and \(\alpha_{P}\), \(\alpha_{Q}\), and \(\alpha_{R}\) are the specific rotations of P, Q, and R, respectively.

Solve for specific rotations in the second solution

In the second solution, we have 40% R, 35% Q, and 25% P and its specific rotation is -8.5°. We can write: $$\alpha_{2} = \alpha_{P}(0.25) + \alpha_{Q}(0.35) + \alpha_{R}(0.4)$$ $$-8.5 = 0.25\alpha_{P} + 0.35\alpha_{Q} + 0.4\alpha_{R}$$ Where \(\alpha_{2}\) is the specific rotation of the second solution.

Solve the system of equations

Now, we have two equations with three unknowns: 1) $$-5 = 0.4\alpha_{P} + 0.1\alpha_{Q} + 0.5\alpha_{R}$$ 2) $$-8.5 = 0.25\alpha_{P} + 0.35\alpha_{Q} + 0.4\alpha_{R}$$ We cannot solve this system directly for each unknown, but we can eliminate one of the variables by multiplying the first equation by 2.5 and subtracting the second equation: $$(-5 \times 2.5) - (-8.5) = (0.4\alpha_{P} \times 2.5) - 0.25\alpha_{P} + (0.1\alpha_{Q} \times 2.5) - 0.35\alpha_{Q}$$

Determine the unknown specific rotation

Solving the above equation, we get: $$-12.5 + 8.5 = \alpha_{P} + 0.25\alpha_{Q}$$ $$-4 = \alpha_{P} + 0.25\alpha_{Q}$$ Now, we can substitute this equation into either equation 1) or 2). Choosing equation 1), we have: $$-5 = 0.4(-4 - 0.25\alpha_{Q}) + 0.1\alpha_{Q} + 0.5\alpha_{R}$$ $$-1 = -0.15\alpha_{Q} + 0.5\alpha_{R}$$ $$\alpha_{R} = 3\alpha_{Q} - 2$$ Now by substituting the determined relationship between \(\alpha_{Q}\) and \(\alpha_{R}\) in equation 2) we can solve for either. $$-8.5 = 0.25(-4 - 0.25\alpha_{Q}) + 0.35\alpha_{Q} + 0.4(3\alpha_{Q} - 2)$$ $$-7 = \alpha_{Q}$$ Now we can find \(\alpha_{R}\) from the relationship we found earlier: $$\alpha_{R} = 3(-7) - 2$$ $$\alpha_{R} = -23$$ Finally, we can find \(\alpha_{P}\) from the equation we found in step 4: $$-4 = \alpha_{P} + 0.25(-7)$$ $$\alpha_{P} = 13$$ We have determined that \(\alpha_P = 13°\), \(\alpha_Q = -7°\), and \(\alpha_R = -23°\).

Check the correct statement

Now we can compare our results with the given statements: (A) P is levorotatory with -13° specific angle of rotation: This statement is incorrect because \(\alpha_{P} = 13°\), P is dextrorotatory. (B) R is levorotatory with -18° specific angle of rotation: This statement is incorrect because \(\alpha_{R} = -23°\). (C) Q is levorotatory with -13° specific angle of rotation: This statement is incorrect because \(\alpha_{Q} = -7°\). (D) Magnitude of the specific angle of rotation for P and Q is 18°: This statement is correct because \(|13 - (-7)| = 18°\).

Key concepts

Optical Isomerism

Optical isomerism is a form of stereoisomerism related to the ability of molecules to rotate plane-polarized light. This characteristic occurs due to the presence of chiral centers, which are carbon atoms bonded to four different groups. As these molecules interact with polarized light, they rotate the light in different directions depending on their arrangement. This rotation can be measured and is unique to each optical isomer or enantiomer. Understanding optical isomerism is crucial in various fields, including pharmaceuticals, as the two enantiomers of a compound can have significantly different biological effects.

• Optical activity depends on chiral centers.
• Chiral molecules rotate plane-polarized light.
• Different isomers can have distinct bioactivities.

Enantiomers

Enantiomers are a type of stereoisomer that are non-superimposable mirror images of each other. They possess identical physical properties, such as melting point and solubility, but differ in how they interact with polarized light and other chiral environments. One enantiomer will rotate light in a clockwise direction, while the other will do so counterclockwise. When a mixture contains equal amounts of both enantiomers, it is referred to as a racemic mixture and tends to show no net optical rotation.

• Enantiomers are mirror images.
• They have identical properties except optical activity.
• They differ in biological interactions.

Levorotatory and Dextrorotatory

Molecules that cause the rotation of plane-polarized light to the left are termed levorotatory, while those rotating it to the right are dextrorotatory. This property is denoted by a negative or positive sign, respectively, preceding the specific rotation value. For example, \(\alpha_{P} = 13^{\circ}\) means compound P is dextrorotatory. Whether a compound is levorotatory or dextrorotatory depends on its stereochemistry, which can influence its biological behavior.

• Levorotatory: Rotates light to the left, indicated by "-".
• Dextrorotatory: Rotates light to the right, indicated by "+".
• Determined by examining optical rotation.

Polarimetry

Polarimetry is a technique to measure the angle of rotation caused by passing polarized light through an optically active substance. It involves using a polarimeter, a device that contains a light source, an analyzer, and a sample cell. By measuring how much the light is rotated, scientists can identify and quantify enantiomeric excess in solutions. The specific rotation, denoted as \(\alpha\), is dependent on the wavelength of light, temperature, and concentration of the substance. Polarimetry plays a key role in fields like chemistry and pharmacology.

• Polarimeter measures optical rotation.
• Important for analyzing optical purity.
• Specific rotation is a key measurement.

System of Equations in Chemistry

In chemistry, systems of equations can be used to solve problems involving mixtures of substances, like determining the specific rotations of different compounds in a solution. By creating equations based on known concentrations and observed rotations, it's possible to solve for unknown values, such as the specific rotation of each component. This method was employed in the original solution to calculate the specific rotations for isomers P, Q, and R. It shows the power of mathematics in quantitatively understanding chemical phenomena.

• Equations model mixtures and reactions.
• Used to find unknown properties.
• Highlights math's role in chemistry.