4. A sample of the amino acid cystine has a specific rotation of -68°. Pure (-)-cystine has a specific rotation of -339°. What are the mole percentages of the levorotatory and dextrorotatory enantiomers in the sample? NH₂ OH HO NH₂ cystine
4. A sample of the amino acid cystine has a specific rotation of -68°. Pure (-)-cystine has a specific rotation of -339°. What are the mole percentages of the levorotatory and dextrorotatory enantiomers in the sample? NH₂ OH HO NH₂ cystine
Chemistry
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ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
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Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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![**Problem Statement:**
**4. A sample of the amino acid cystine has a specific rotation of -68°. Pure (-)-cystine has a specific rotation of -339°. What are the mole percentages of the levorotatory and dextrorotatory enantiomers in the sample?**
**Description of Image:**
The image also includes a structural diagram of the amino acid cystine. The structure shows two cysteine molecules linked by a disulfide bond (S-S). Each cysteine contains an amino group (NH2), a carboxyl group (COOH), and a sulfhydryl group (SH), but the SH groups are connected by the disulfide bond.
**Solution:**
To find the mole percentages of the levorotatory and dextrorotatory enantiomers, you can use the formula for specific rotation:
\[ [\alpha]_{\text{observed}} = f_1 [\alpha]_{\text{pure} (-)} + f_2 [\alpha]_{\text{pure} (+)} \]
Where:
- \([\alpha]_{\text{observed}}\) is the observed specific rotation.
- \([\alpha]_{\text{pure} (-)}\) is the specific rotation of pure (-)-cystine.
- \([\alpha]_{\text{pure} (+)}\) is the specific rotation of pure (+)-cystine, which is equal and opposite to that of the pure (-)-cystine.
- \(f_1\) and \(f_2\) are the mole fractions of the (-)- and (+)-enantiomers, respectively.
Given that \([\alpha]_{\text{observed}} = -68°\), \([\alpha]_{\text{pure} (-)} = -339°\), and \([\alpha]_{\text{pure} (+)} = +339°\):
1. Since \(f_1 + f_2 = 1\), let \(f_1 = x\) and \(f_2 = 1 - x\).
2. Substituting the values into the formula:
\[ -68 = x(-339) + (1 - x)(339) \]
3. Simplify and solve for \(x\):
\[ -68 = -339x + 339 - 339x \]
\[ -68 = 339 - 678x \]
\[ -68](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa44093f9-865e-42e2-9761-7238b47b32ab%2F45a3030d-d31d-45c8-829d-46d774215a53%2F3iy3vm9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
**4. A sample of the amino acid cystine has a specific rotation of -68°. Pure (-)-cystine has a specific rotation of -339°. What are the mole percentages of the levorotatory and dextrorotatory enantiomers in the sample?**
**Description of Image:**
The image also includes a structural diagram of the amino acid cystine. The structure shows two cysteine molecules linked by a disulfide bond (S-S). Each cysteine contains an amino group (NH2), a carboxyl group (COOH), and a sulfhydryl group (SH), but the SH groups are connected by the disulfide bond.
**Solution:**
To find the mole percentages of the levorotatory and dextrorotatory enantiomers, you can use the formula for specific rotation:
\[ [\alpha]_{\text{observed}} = f_1 [\alpha]_{\text{pure} (-)} + f_2 [\alpha]_{\text{pure} (+)} \]
Where:
- \([\alpha]_{\text{observed}}\) is the observed specific rotation.
- \([\alpha]_{\text{pure} (-)}\) is the specific rotation of pure (-)-cystine.
- \([\alpha]_{\text{pure} (+)}\) is the specific rotation of pure (+)-cystine, which is equal and opposite to that of the pure (-)-cystine.
- \(f_1\) and \(f_2\) are the mole fractions of the (-)- and (+)-enantiomers, respectively.
Given that \([\alpha]_{\text{observed}} = -68°\), \([\alpha]_{\text{pure} (-)} = -339°\), and \([\alpha]_{\text{pure} (+)} = +339°\):
1. Since \(f_1 + f_2 = 1\), let \(f_1 = x\) and \(f_2 = 1 - x\).
2. Substituting the values into the formula:
\[ -68 = x(-339) + (1 - x)(339) \]
3. Simplify and solve for \(x\):
\[ -68 = -339x + 339 - 339x \]
\[ -68 = 339 - 678x \]
\[ -68
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