Ch. 15 - Structural Identification II: Nuclear Magnetic Resonance Spectroscopy

Mullins - Organic Chemistry: A Learner Centered Approach 1st Edition

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Mullins 1st Edition

Ch. 15 - Structural Identification II: Nuclear Magnetic Resonance Spectroscopy

Problem 4

Chapter 14, Problem 4

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

Radical chlorination, a reaction studied in Chapters 5 and 11, replaces a hydrogen with a chlorine. Assuming there is no regioselectivity (all hydrogens can react equally), how many different chloroalkanes are possible upon reaction of each alkane with one equivalent of Cl₂

(a)

(b)

(c)

(d)

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Textbook Question

(a) Rank the following bonds in terms of the strength of their bond dipole (1 = weakest, 6 = strongest).

(b) Which carbon has the largest δ⁺ ?

C―F, C―Br, C―I, C―H, C―C, C―Cl

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Textbook Question

The spectrum shown here is for a molecule with a molecular formula of C₄H₉Cl. Suggest a structure for this molecule.

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1038

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Textbook Question

Draw the peak that would correspond to the hydrogens at C₃ in the molecule shown. Be sure to indicate the integration and place it at an appropriate chemical shift.

841

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Textbook Question

Calculate the index of hydrogen deficiency for curacin A, which has a molecular formula of C₂₃H₃₅NOS .

905

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