Table of contents

1. A Review of General Chemistry5h 5m
2. Molecular Representations1h 14m
3. Acids and Bases2h 46m
4. Alkanes and Cycloalkanes4h 17m
5. Chirality3h 39m
6. Thermodynamics and Kinetics1h 22m
7. Substitution Reactions1h 48m
8. Elimination Reactions2h 30m
9. Alkenes and Alkynes2h 9m
10. Addition Reactions3h 19m
11. Radical Reactions1h 58m
12. Alcohols, Ethers, Epoxides and Thiols2h 42m
13. Alcohols and Carbonyl Compounds2h 17m
14. Synthetic Techniques1h 26m
15. Analytical Techniques:IR, NMR, Mass Spect7h 3m
16. Conjugated Systems6h 13m
17. Ultraviolet Spectroscopy51m
18. Aromaticity2h 34m
19. Reactions of Aromatics: EAS and Beyond5h 1m
20. Phenols55m
21. Aldehydes and Ketones: Nucleophilic Addition4h 56m
22. Carboxylic Acid Derivatives: NAS2h 51m
23. The Chemistry of Thioesters, Phophate Ester and Phosphate Anhydrides1h 10m
24. Enolate Chemistry: Reactions at the Alpha-Carbon1h 53m
25. Condensation Chemistry2h 9m
26. Amines1h 43m
27. Heterocycles2h 0m
28. Carbohydrates5h 53m
29. Amino Acids4h 20m
30. Peptides and Proteins2h 42m
31. Catalysis in Organic Reactions1h 30m
32. Lipids 2h 50m
33. The Organic Chemistry of Metabolic Pathways2h 52m
34. Nucleic Acids1h 32m
35. Transition Metals6h 14m
36. Synthetic Polymers1h 49m

15. Analytical Techniques:IR, NMR, Mass Spect

1H NMR:Spin-Splitting (N + 1) Rule

Organic Chemistry

15. Analytical Techniques:IR, NMR, Mass Spect

1H NMR:Spin-Splitting (N + 1) Rule

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2:55 minutes

Problem 4

Textbook Question

Every number in Pascal’s triangle is the sum of the two numbers above it. Given this, fill in the missing numbers.

Verified step by step guidance

Identify the pattern in Pascal's triangle: each number is the sum of the two numbers directly above it.

To find the value of (a), add the two numbers above it: 1 and 2. So, (a) = 1 + 2.

To find the value of (b), add the two numbers above it: 3 and (a). So, (b) = 3 + (a).

To find the value of (c), add the two numbers above it: (a) and 1. So, (c) = (a) + 1.

To find the value of (d), add the two numbers above it: 1 and 4. So, (d) = 1 + 4. To find the value of (e), add the two numbers above it: 10 and 5. So, (e) = 10 + 5.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Pascal's Triangle

Pascal's Triangle is a triangular array of binomial coefficients, where each number is the sum of the two directly above it. The triangle starts with a '1' at the top, and each subsequent row corresponds to the coefficients of the binomial expansion. Understanding its structure is essential for solving problems related to combinatorics and algebra.

Binomial Coefficients

Binomial coefficients are the numbers that appear in Pascal's Triangle and represent the coefficients in the expansion of a binomial expression (a + b)^n. They are denoted as C(n, k) or 'n choose k', indicating the number of ways to choose k elements from a set of n elements. This concept is crucial for understanding the relationships between the numbers in the triangle.

Recursion in Mathematics

Recursion in mathematics refers to defining a sequence or function in terms of itself. In the context of Pascal's Triangle, each entry can be calculated using the recursive formula: C(n, k) = C(n-1, k-1) + C(n-1, k). This principle allows for the systematic filling of the triangle and is fundamental for solving problems that involve patterns and sequences.

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Related Practice

Textbook Question

Predict the splitting patterns for the signals given by the compounds in Problem 4.

c. CH₂=CCl₂

Textbook Question

Label each set of chemically equivalent protons, using a for the set that will be at the lowest frequency in the ¹H NMR spectrum, b for the next lowest, and so on. Indicate the multiplicity of each signal.

Textbook Question

In Figure 15.34, H_a was assumed to be 'up.' How does the analysis change if we assume instead that H_a is down?

Textbook Question

Though Figure 15.34 was concerned with the appearance of H_a, how would H_b appear in the spectrum?

Textbook Question

Indicate the number of signals and the multiplicity of each signal in the 1H NMR spectrum of each of the following compounds:

c. ClCH₂CH₂CH₂Cl

Textbook Question

Indicate the number of signals and the multiplicity of each signal in the 1H NMR spectrum of each of the following compounds:

a. CH₃CH₂CH₂CH₂CH₂CH₃

Textbook Question

Predict the splitting patterns for the signals given by the compounds in Problem 4.

Textbook Question

Predict the splitting patterns for the signals given by the compounds in Problem 4.

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