Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
Question
Chapter 20, Problem 20.68E
Interpretation Introduction

(a)

Interpretation:

The activation energy for the isomerization reaction is to be predicted.

Concept introduction:

A small packet of energy is known as quanta. Light is emitted in the form of quanta or photons. The Planck’s law gives the relation between the energy and wavelength, frequency and wavenumber.

Expert Solution
Check Mark

Answer to Problem 20.68E

The activation energy for the isomerization reaction is 159.6kJmol1.

Explanation of Solution

The given isomerization reaction is,

hv+cis-retinaltrans-retinal

It is given that the wavelength of least energetic photon is 750nm.

To calculate the activation energy of given isomerization reaction the formula used is,

E=hcλ

Where,

h is the Planck’s constant

E is the energy

c is the speed of light

λ is the wavelength

Substitute the values of Planck’s constant, speed of light and wavelength of photon in the given formula.

E=6.626×1034Js×3×108ms1750×109mE=2.65×1019J

Thus, the energy for theleast energetic photon is 2.65×1019J. The activation energy for the isomerization reaction is calculated for one mole of photons as shown below.

EA=2.65×1019J×6.022×1023mol1EA=159.6×103Jmol1EA=159.6kJmol1

Thus, the activation energy for the isomerization reaction is 159.6kJmol1.

Conclusion

The activation energy for the isomerization reaction is 159.6kJmol1.

Interpretation Introduction

(b)

Interpretation:

The value of pre-exponential factor for the given isomerization reaction is to be calculated.

Concept introduction:

The Arrhenius equation gives the temperature dependence of reaction rates.

k=AeEA/RT

Where,

k is the rate constant.

A is the pre-exponential factor.

EA is the activation energy.

R is the gas constant and its value is 8.314JK1mol1.

T is the temperature.

The pre-exponential factor is also known as the frequency factor or the steric factor.

Expert Solution
Check Mark

Answer to Problem 20.68E

The value of pre-exponential factor is 2.347×1038s1.

Explanation of Solution

The rate constant for the given isomerization reaction is 3×1011s1 at 37°C. The activation energy for the given reaction is 159.6kJmol1.

The Arrhenius equation can be used for the calculation of activation energy. The Arrhenius equation is,

k=AeEA/RT

Where,

k is the rate constant.

A is the pre-exponential factor.

EA is the activation energy.

R is the gas constant and its value is 8.314JK1mol1.

T is the temperature.

Conversion of temperature in Celsius to Kelvin is done by the formula,

T(K)=T(°C)+273K

Substitute the temperature (37°C) in the formula.

T=37°C+273T=310K

Thus, the given temperature in Kelvin is 310K.

Substitute the values of activation energy, gas constant, rate constant and temperature.

3×1011s1=Ae159.6kJ/mol/8.314JK1mol1×310K3×1011s1=Ae159.6×103J/mol/8.314JK1mol1×310K3×1011s1=Ae61.92

The above equation if further solved to obtain the value of pre-exponential factor as shown below.

3×1011s1=A(1.278×1027)A=3×1011s11.278×1027A=2.347×1038s1

The value of pre-exponential factor is 2.347×1038s1.

Conclusion

The value of pre-exponential factor is 2.347×1038s1.

Interpretation Introduction

(c)

Interpretation:

The value of the rate constant at 2°C for the given isomerization reaction taking place in ice fish eye is to be predicted.

Concept introduction:

The Arrhenius equation gives the temperature dependence of reaction rates.

k=AeEA/RT

Where,

k is the rate constant.

A is the pre-exponential factor.

EA is the activation energy.

R is the gas constant and its value is 8.314JK1mol1.

T is the temperature.

The pre-exponential factor is also known as the frequency factor or the steric factor.

Expert Solution
Check Mark

Answer to Problem 20.68E

The value of the rate constant at 2°C for the given isomerization reaction taking place in ice fish eye is 4.04×107s1.

Explanation of Solution

It is given that the rate constant at 37°C(T1) is 3×1011s1 and value of activation energy is 159.6kJmol1.

The form of Arrhenius equation used to calculate the rate constant at different temperature is,

lnk1k2=(EAR)(1T11T2) …(1)

Where,

k1 and k2 is the rate constant at two different temperatures.

EA is the activation energy.

R is the gas constant and its value is 8.314JK1mol1.

T1 and T2 is the are different temperatures.

Conversion of temperature in Celsius to Kelvin is done by the formula,

T(K)=T(°C)+273K

Substitute the temperature (2°C) in the formula.

T=2°C+273T=271K

Thus, the temperature (T2) is 271K.

Substitute the values of activation energy, temperatures, the rate constant at 310K and gas constant in equation (1).

ln3×1011s1k2=(159.6×103J/mol8.314JK1mol1)(1310K1271K)ln3×1011s1k2=19196.5K(39K271K×310K)ln3×1011s1k2=8.912

Take inverse of logarithm on both sides of equation to solve for the value of rate constant at 271K as shown below.

3×1011s1k2=e8.9123×1011s1k2=7420.487k2=3×1011s17420.487k2=4.04×107s1

Thus, the value of rate constant at 271K is 4.04×107s1.

Conclusion

The value of the rate constant at 2°C for the given isomerization reaction taking place in ice fish eye is 4.04×107s1.

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Chapter 20 Solutions

Physical Chemistry

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