# The force constant for CO bond needs to be determined. Concept Introduction: The force constant may be defined as the restoring force per unit distance by which the bond is stretched out. The value of force constant is used for the analysis and characteristics of the molecules. It is also used for the calculations of other constants associated with the analysis.

### Principles of Instrumental Analysis

7th Edition
Douglas A. Skoog + 2 others
Publisher: Cengage Learning
ISBN: 9781305577213

### Principles of Instrumental Analysis

7th Edition
Douglas A. Skoog + 2 others
Publisher: Cengage Learning
ISBN: 9781305577213

#### Solutions

Chapter 16, Problem 16.1QAP

(a)

Interpretation Introduction

## Interpretation: The force constant for CO bond needs to be determined.Concept Introduction: The force constant may be defined as the restoring force per unit distance by which the bond is stretched out. The value of force constant is used for the analysis and characteristics of the molecules. It is also used for the calculations of other constants associated with the analysis.

Expert Solution

The force constant for CO bond is 1.895×103kg/s2 .

### Explanation of Solution

The expression for the force constant is:

k=(ν¯×2πc)2μ ...... (I)

Here, the velocity of light is c , the wave number of absorption peak is ν¯ , and the reduced mass is μ .

The expression for the reduced mass is:

μ=m1m2m1+m2

Here, mass of the carbon atom is m1 and the mass of the oxygen atom is m2 .

Substitute m1m2m1+m2 for μ in Equation (I).

k=(ν¯×2πc)2(m1m2m1+m2) ...... (II)

Substitute 2170cm1 for ν¯ , 3×1010cm/s for c , 12g/mol for m1 , 16g/mol for m2 , and 3.14 for π in Equation (II).

k=(( 2170 cm 1 )( 2)( 3.14)( 3× 10 10 cm/s ))2( ( 12g/ mol )( 16g/ mol ) ( 12g/ mol )+( 16g/ mol ))=(4.088× 10 14 s 1)2( ( 12( g/ mol )×( 1kg 1000g ) )( 16( g/ mol )×( 1kg 1000g ) ) ( 12( g/ mol )×( 1kg 1000g ) )+( 16( g/ mol )×( 1kg 1000g ) ))=(1.66× 10 29s 2)( ( 0.012 kg/ mol )( 0.016 kg/ mol ) ( 0.012 kg/ mol )+( 0.016 kg/ mol ))=(1.66× 10 29s 2)(6.857× 10 3kg/mol)

k=(1.66× 10 29s 2)(6.857× 10 3( kg/ mol )×( 1mol 6× 10 23 atom ×1atom))=(1.66× 10 29s 2)(1.142× 10 26kg)=1895.72kg/s2=1.895×103kg/s2

Thus, the force constant for CO bond is 1.895×103kg/s2 .

(b)

Interpretation Introduction

### Interpretation: The corresponding wave number at the peak for C14O needs to be determined.Concept Introduction: The wave number may be defined as the wavelength divide by the unit distance. The wave number is generally expressed in cm−1 .

Expert Solution

The corresponding wave number at the peak for C14O is 2155.7cm1 .

### Explanation of Solution

The expression for the wave number is:

ν¯=12πckμ ...... (III)

Substitute m1m2m1+m2 for μ in Equation (III).

ν¯=12πck( m 1 m 2 m 1 + m 2 ) ...... (IV)

Substitute 3×1010cm/s for c , 12g/mol for m1 , 16g/mol for m2 , 1.895×103kg/s2 for k , and 3.14 for π in Equation (II).

ν ¯ = 1 ( 2 )( 3.14 )( 3× 10 10 cm/s ) ( 1.895× 10 3 kg/s 2 ) ( ( 12g/ mol )( 16g/ mol ) ( 12g/ mol )+( 16g/ mol ) )

=( 5.307× 10 12 s/ cm )( ( 1.895× 10 3 kg/s 2 ) ( 12( g/ mol )×( 1kg 1000g ) )( 16( g/ mol )×( 1kg 1000g ) ) ( 12( g/ mol )×( 1kg 1000g ) )+( 16( g/ mol )×( 1kg 1000g ) ) )

=( 5.307× 10 12 s/ cm )( ( 1.895× 10 3 kg/s 2 ) ( 0.012 kg/ mol )( 0.016 kg/ mol ) ( 0.012 kg/ mol )+( 0.016 kg/ mol ) )

=( 5.307× 10 12 s/ cm )( ( 1.895× 10 3 kg/s 2 ) 6.857× 10 3 kg/ mol )

ν¯=(5.307× 10 12s/cm)( ( 1.895× 10 3 kg/s 2 ) 6.857× 10 3 kg/ mol )=(5.307× 10 12s/cm)( ( 1.895× 10 3 kg/s 2 ) ( 6.857× 10 3 ( kg/ mol )×( 1mol 6× 10 23 atom ×1atom ) ) )=(5.307× 10 12s/cm)( ( 1.895× 10 3 kg/s 2 ) ( 1.142× 10 26 kg ) )=(5.307× 10 12s/cm)1.65× 10 29 /s 2

ν¯=(5.307× 10 12s/cm)(4.062× 10 14/s)=2155.7cm1

Thus, the corresponding wave number at the peak for C14O is 2155.7cm1 .

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