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Chapter 1: Problem 106

The origin of the fundamental aborption band in the vibration-rotation spectrum of \({ }^{1} \mathrm{H}^{35} \mathrm{Cl}\) lies at wavenumber \(\hat{v}=2886 \mathrm{~cm}^{-1} .\) Calculate the corresponding (i) frequency, (ii) wavelength, and (iii) energy in units of eV and kJ mol \(^{-1}\).

Short Answer

Expert verified
Frequency: \(8.658 \times 10^{13} \text{ s}^{-1}\), Wavelength: \(3.46 \times 10^{-4} \text{ cm}\), Energy: 0.358 eV, 34.53 kJ/mol.

Step by step solution

01

Convert Wavenumber to Frequency

The relationship between wavenumber \( \hat{v} \) and frequency \( u \) is given by \( u = c \cdot \hat{v} \), where \( c \) is the speed of light \(3 \times 10^{10} \text{ cm/s} \). For \( \hat{v} = 2886 \text{ cm}^{-1} \), calculate the frequency: \[ u = 3 \times 10^{10} \text{ cm/s} \times 2886 \text{ cm}^{-1} = 8.658 \times 10^{13} \text{ s}^{-1} \].
02

Calculate Wavelength

The wavelength \( \lambda \) is related to the speed of light \( c \) and frequency \( u \) by the formula \( \lambda = \frac{c}{u} \). Substitute the speed of light \(3 \times 10^{10} \text{ cm/s}\) and the frequency calculated in Step 1: \[ \lambda = \frac{3 \times 10^{10} \text{ cm/s}}{8.658 \times 10^{13} \text{ s}^{-1}} \approx 3.46 \times 10^{-4} \text{ cm} \].
03

Convert Frequency to Energy (eV)

To convert frequency to energy in electron volts (eV), use the formula \( E = h u \), where \( h \) (Planck's constant) is \( 4.136 \times 10^{-15} \text{ eV s} \). Calculate the energy: \[ E = 8.658 \times 10^{13} \text{ s}^{-1} \times 4.136 \times 10^{-15} \text{ eV s} \approx 0.358 \text{ eV} \].
04

Convert Energy to kJ/mol

Energy in kJ/mol can be found using the formula \( E_{\text{kJ/mol}} = E_{\text{eV}} \times 96.485 \), where the conversion factor 96.485 is used to convert eV to kJ/mol. Thus: \[ E_{\text{kJ/mol}} = 0.358 \text{ eV} \times 96.485 \approx 34.53 \text{ kJ/mol} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wavenumber
The concept of wavenumber is fundamental in understanding the vibration-rotation spectrum of molecules like \({ }^{1} \mathrm{H}^{35} \mathrm{Cl}\). Wavenumber is the measure of the number of wave cycles in one unit of distance. It is usually expressed in units of inverse centimeters (cm\(^{-1}\)). This makes it a convenient unit for describing molecular vibrations and rotations.
  • Wavenumber is inversely proportional to wavelength, meaning that higher wavenumbers correspond to shorter wavelengths.
  • It is a key parameter in the field of spectroscopy, as it helps identify energy transitions occurring at specific frequencies.
For \({ }^{1} \mathrm{H}^{35} \mathrm{Cl}\), the fundamental absorption band is at a wavenumber of 2886 cm\(^{-1}\), which sets the stage for frequency and energy calculations.
Frequency Calculation
Frequency, often denoted as \( u \), is a measure of how often the peaks of a wave pass a point in one second. It is a crucial concept in the vibrations and rotations of molecules, as it defines the energy involved.
  • The formula relating wavenumber and frequency is \( u = c \cdot \hat{v} \), where \( c \) is the speed of light.
  • The speed of light \( c \) is approximately \( 3 \times 10^{10} \text{ cm/s} \).
In this exercise, using \( \hat{v} = 2886 \text{ cm}^{-1} \) results in a frequency of \( 8.658 \times 10^{13} \text{ s}^{-1} \). This frequency directly relates to the energy emitted or absorbed by the \({ }^{1} \mathrm{H}^{35} \mathrm{Cl}\) molecule.
Wavelength Determination
Wavelength \( \lambda \) is the distance between successive peaks of a wave. It is crucial in understanding the nature of light and electromagnetic waves.
  • Wavelength is inversely proportional to frequency, shown by the equation \( \lambda = \frac{c}{u} \).
  • For light waves, the speed of light \( c \) is constant.
With the frequency \( 8.658 \times 10^{13} \text{ s}^{-1} \) calculated previously, the wavelength is found to be approximately \( 3.46 \times 10^{-4} \text{ cm} \).
This short wavelength indicates a high-energy transition in the vibration-rotation spectrum of \({ }^{1} \mathrm{H}^{35} \mathrm{Cl}\).
Energy Conversion
The energy of a wave can be determined from its frequency, using Planck's law. Energy is expressed in units like electron volts (eV) and kilojoules per mole (kJ/mol).
  • The formula \( E = h u \) computes energy from frequency, where Planck's constant \( h = 4.136 \times 10^{-15} \text{ eV s} \).
  • This has been used to calculate the energy of the \({ }^{1} \mathrm{H}^{35} \mathrm{Cl}\) absorption as about \( 0.358 \text{ eV} \).
  • To convert to kJ/mol, multiply by the conversion factor 96.485, resulting in about \( 34.53 \text{ kJ/mol} \).
These conversions are vital when comparing the energies across various spectroscopic lines or when discussing the energy requirements of molecular interactions.
Planck's Constant
Planck's constant \( h \) is pivotal in linking the energy of electromagnetic waves to their frequency. It is a fundamental constant in quantum mechanics and has a value of \( 6.626 \times 10^{-34} \text{ J s} \) or \( 4.136 \times 10^{-15} \text{ eV s} \) when expressed in electron volts.
  • Planck's constant enables the conversion between frequency and energy via the relation \( E = h u \).
  • It underpins many theories and principles in quantum physics, including the photoelectric effect.
  • In spectroscopy, knowing \( h \) allows us to determine the energy transitions within molecules based on observed frequencies.
Understanding Planck鈥檚 constant is essential for grasping how energy quantization occurs in physical systems, especially in molecular vibrational and rotational motions.

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Most popular questions from this chapter

Simplify by factorization and cancellation:\(\frac{3}{18}\)

Simplify if possible:\(a^{3} / a^{2}\)

The vapour pressure of water at \(20^{\circ} \mathrm{C}\) is recorded as \(p\left(\mathrm{H}_{2} \mathrm{O}, 20^{\circ} \mathrm{C}\right)=17.5\) Torr. Express this in terms of (i) the base SI unit of pressure, (ii) bar, (iii) atm.

Simplify if possible:\(a^{3} a^{-4}\)

Evaluate:\(7+3 \times 4-5\)
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