Atkins' Physical Chemistry
Atkins' Physical Chemistry
11th Edition
ISBN: 9780198769866
Author: ATKINS, P. W. (peter William), De Paula, Julio, Keeler, JAMES
Publisher: Oxford University Press
Question
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Chapter 7, Problem 7D.12P
Interpretation Introduction

Interpretation:

The expression for the transmission probability has to be derived.  The expression for the condition when transmission probability reduces to T16ε(1ε)eκW when κW is much higher than 1 has to be shown.

Concept introduction:

In quantum mechanics, the wavefunction is given by ψ.  The wavefunction contains all the information about the state of the system.  The wavefunction is the function of the coordinates of particles and time.

Expert Solution & Answer
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Answer to Problem 7D.12P

The expression for the transmission probability is shown below.

    T=((eκWeκW)216ε(1ε)+1)1

When κW1, then the expression for the transmission probability is shown below.

    T=16ε(1ε)e2κW

Explanation of Solution

The transmission probability is given by the expression shown below.

    T=|A'|2|A|2                                                                                                        (1)

Where,

  • A and A' are the coefficients.

At the edge of barrier (x=0), the relationship between the coefficients A, B, C, and D is given by the expression as shown below.

    A+B=C+D                                                                                               (2)

At the edge of barrier (x=W), the relationship between the coefficients A', C, and D is given by the expression as shown below.

    CeκW+DeκW=A'eikW                                                                                 (3)

Where,

  • W is the potential energy width.
  • κ is a constant.

At the edge of barrier (x=0), the slope is given by the expression as shown below.

    ikAikB=κCκD                                                                                      (4)

Where,

  • k is a constant.

At the edge of barrier (x=0), the slope is given by the expression as shown below.    κCeκWκDeκW=ikA'eikW                                                                          (5)

Rearrange the equation (2) for the value of B.

    B=C+DA                                                                                               (6)

Rearrange the equation (4) for the value of B.

    B=ikAκC+κDik                                                                                        (7)

Substitute the value of equation (6) in equation (7).

    C+DA=ikAκC+κDikC+DA=AκCik+κDik                                                                           (8)

Rearrange the equation (8) for the value of C.

    C=2Aik+D(κik)κ+ik                                                                                   (9)

Rearrange the equation (3) for the value of A'.

    A'=CeκW+DeκWeikWA'=(CeκW+DeκW)eikW                                                                           (10)

Rearrange the equation (5) for the value of A'.

    A'=κeikW(CeκWDeκW)ik                                                                        (11)

Substitute the value of equation (10) in equation (11).

    (CeκW+DeκW)eikW=κeikW(CeκWDeκW)ik                                          (12)

Rearrange the equation (12) for the value of C.

  C=(κik+1)De2κWκik1C=(κ+ik)De2κWκik                                                                                    (13)

Substitute the value of equation (9) in the equation (13).

    (κ+ik)De2κWκik=2Aik+D(κik)κ+ik                                                           (14)

Rearrange the equation (14) for the value of D.

    D=2Aik(κik)(κ+ik)2e2κW(κik)2                                                                    (15)

Substitute the value of D from equation (15) to the equation (13).

    C=(κ+ik)(2Aik(κik)(κ+ik)2e2κW(κik)2)e2κWκikC=2Aik(κ+ik)e2κW(κ+ik)2e2κW(κik)2                                          (16)

Substitute the values of D from equation (15) and C from equation (16) to the equation (10).

    A'=((2Aik(κ+ik)e2κW(κ+ik)2e2κW(κik)2)eκW+(2Aik(κik)(κ+ik)2e2κW(κik)2)eκW)eikWA'=2Aik(κ+ik)2e2κW(κik)2((κ+ik)eκW+(κik)eκW)eikWA'=4AiκkeκWeikW(κ+ik)2e2κW(κik)2A'=4AiκkeikW(κ+ik)2eκW(κik)2eκW

Substitute the value of A' from above equation in the equation (1).

    T=|4AiκkeikW(κ+ik)2eκW(κik)2eκW|2|A|2=(4iκkeikW(κ+ik)2eκW(κik)2eκW)(4iκkeikW(κ+ik)2eκW(κik)2eκW)=16i2κ2k2((κ+ik)2eκW(κik)2eκW)2

Substitute the value i2=1 on the above equation.

    T=16(1)κ2k2((κ+ik)2eκW(κik)2eκW)2=16κ2k2((κ+ik)2eκW(κik)2eκW)2

The above equation is further solved to get the expression shown below.

    T=16κ2k2(κ2+k2)2(eκWeκW)2+16κ2k2=((κ2+k2)2(eκWeκW)2+16κ2k216κ2k2)1=((κ2+k2)2(eκWeκW)216κ2k2+1)1                                                 (17)

The constants κ and k can be written in the terms of energy as shown below.

    κ=[2m(V0E)]1/2k=[2mE]1/2

Where,

  • E is the total energy.
  • V is the potential energy.
  • m is the mass.
  • is a constant.

Substitute the value of κ and k in the equation (17).

    T=((([2m(V0E)]1/2)2+([2mE]1/2)2)2(eκWeκW)216([2m(V0E)]1/2)2([2mE]1/2)2+1)1

The above equation is further simplified to get an expression shown below.

    T=((V02)(eκWeκW)216E(V0E)+1)1T=((eκWeκW)216EV0(1EV0)+1)1                                                                    (20)

The ratio of total energy and potential energy is given by the expression shown below.

    ε=EV0

Substitute the value of EV0 in the equation (20).

  T=((eκWeκW)216ε(1ε)+1)1

When κW1, then the eκW term is very negligible and the eκW term is very large in comparison to 1.  Therefore, the above expression can be simplified as shown below.

    T=((eκW)216ε(1ε))1=(e2κW16ε(1ε))1=16ε(1ε)e2κW=16ε(1ε)e2κW

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Chapter 7 Solutions

Atkins' Physical Chemistry

Ch. 7 - Prob. 7D.1STCh. 7 - Prob. 7E.1STCh. 7 - Prob. 7E.2STCh. 7 - Prob. 7F.1STCh. 7 - Prob. 7A.1DQCh. 7 - Prob. 7A.2DQCh. 7 - Prob. 7A.3DQCh. 7 - Prob. 7A.4DQCh. 7 - Prob. 7A.1AECh. 7 - Prob. 7A.1BECh. 7 - Prob. 7A.2AECh. 7 - Prob. 7A.2BECh. 7 - Prob. 7A.3AECh. 7 - Prob. 7A.3BECh. 7 - Prob. 7A.4AECh. 7 - Prob. 7A.4BECh. 7 - Prob. 7A.5AECh. 7 - Prob. 7A.5BECh. 7 - Prob. 7A.6AECh. 7 - Prob. 7A.6BECh. 7 - Prob. 7A.7AECh. 7 - Prob. 7A.7BECh. 7 - Prob. 7A.8AECh. 7 - Prob. 7A.8BECh. 7 - Prob. 7A.9AECh. 7 - Prob. 7A.9BECh. 7 - Prob. 7A.10AECh. 7 - Prob. 7A.10BECh. 7 - Prob. 7A.11AECh. 7 - Prob. 7A.11BECh. 7 - Prob. 7A.12AECh. 7 - Prob. 7A.12BECh. 7 - Prob. 7A.13AECh. 7 - Prob. 7A.13BECh. 7 - Prob. 7A.1PCh. 7 - Prob. 7A.2PCh. 7 - Prob. 7A.3PCh. 7 - Prob. 7A.4PCh. 7 - Prob. 7A.5PCh. 7 - Prob. 7A.6PCh. 7 - Prob. 7A.7PCh. 7 - Prob. 7A.8PCh. 7 - Prob. 7A.9PCh. 7 - Prob. 7A.10PCh. 7 - Prob. 7B.1DQCh. 7 - Prob. 7B.2DQCh. 7 - Prob. 7B.3DQCh. 7 - Prob. 7B.1AECh. 7 - Prob. 7B.1BECh. 7 - Prob. 7B.2AECh. 7 - Prob. 7B.2BECh. 7 - Prob. 7B.3AECh. 7 - Prob. 7B.3BECh. 7 - Prob. 7B.4AECh. 7 - Prob. 7B.4BECh. 7 - Prob. 7B.5AECh. 7 - Prob. 7B.5BECh. 7 - Prob. 7B.6AECh. 7 - Prob. 7B.6BECh. 7 - Prob. 7B.7AECh. 7 - Prob. 7B.7BECh. 7 - Prob. 7B.8AECh. 7 - Prob. 7B.8BECh. 7 - Prob. 7B.1PCh. 7 - Prob. 7B.2PCh. 7 - Prob. 7B.3PCh. 7 - Prob. 7B.4PCh. 7 - Prob. 7B.5PCh. 7 - Prob. 7B.7PCh. 7 - Prob. 7B.8PCh. 7 - Prob. 7B.9PCh. 7 - Prob. 7B.11PCh. 7 - Prob. 7C.1DQCh. 7 - Prob. 7C.2DQCh. 7 - Prob. 7C.3DQCh. 7 - Prob. 7C.1AECh. 7 - Prob. 7C.1BECh. 7 - Prob. 7C.2AECh. 7 - Prob. 7C.2BECh. 7 - Prob. 7C.3AECh. 7 - Prob. 7C.3BECh. 7 - Prob. 7C.4AECh. 7 - Prob. 7C.4BECh. 7 - Prob. 7C.5AECh. 7 - Prob. 7C.5BECh. 7 - Prob. 7C.6AECh. 7 - Prob. 7C.6BECh. 7 - Prob. 7C.7AECh. 7 - Prob. 7C.7BECh. 7 - Prob. 7C.8AECh. 7 - Prob. 7C.8BECh. 7 - Prob. 7C.9AECh. 7 - Prob. 7C.9BECh. 7 - Prob. 7C.10AECh. 7 - Prob. 7C.10BECh. 7 - Prob. 7C.1PCh. 7 - Prob. 7C.2PCh. 7 - Prob. 7C.3PCh. 7 - Prob. 7C.4PCh. 7 - Prob. 7C.5PCh. 7 - Prob. 7C.6PCh. 7 - Prob. 7C.7PCh. 7 - Prob. 7C.8PCh. 7 - Prob. 7C.9PCh. 7 - Prob. 7C.11PCh. 7 - Prob. 7C.12PCh. 7 - Prob. 7C.13PCh. 7 - Prob. 7C.14PCh. 7 - Prob. 7C.15PCh. 7 - Prob. 7D.1DQCh. 7 - Prob. 7D.2DQCh. 7 - Prob. 7D.3DQCh. 7 - Prob. 7D.1AECh. 7 - Prob. 7D.1BECh. 7 - Prob. 7D.2AECh. 7 - Prob. 7D.2BECh. 7 - Prob. 7D.3AECh. 7 - Prob. 7D.3BECh. 7 - Prob. 7D.4AECh. 7 - Prob. 7D.4BECh. 7 - Prob. 7D.5AECh. 7 - Prob. 7D.5BECh. 7 - Prob. 7D.6AECh. 7 - Prob. 7D.6BECh. 7 - Prob. 7D.7AECh. 7 - Prob. 7D.7BECh. 7 - Prob. 7D.8AECh. 7 - Prob. 7D.8BECh. 7 - Prob. 7D.9AECh. 7 - Prob. 7D.9BECh. 7 - Prob. 7D.10AECh. 7 - Prob. 7D.10BECh. 7 - Prob. 7D.11AECh. 7 - Prob. 7D.11BECh. 7 - Prob. 7D.12AECh. 7 - Prob. 7D.12BECh. 7 - Prob. 7D.13AECh. 7 - Prob. 7D.13BECh. 7 - Prob. 7D.14AECh. 7 - Prob. 7D.14BECh. 7 - Prob. 7D.15AECh. 7 - Prob. 7D.15BECh. 7 - Prob. 7D.1PCh. 7 - Prob. 7D.2PCh. 7 - Prob. 7D.3PCh. 7 - Prob. 7D.4PCh. 7 - Prob. 7D.5PCh. 7 - Prob. 7D.6PCh. 7 - Prob. 7D.7PCh. 7 - Prob. 7D.8PCh. 7 - Prob. 7D.9PCh. 7 - Prob. 7D.11PCh. 7 - Prob. 7D.12PCh. 7 - Prob. 7D.14PCh. 7 - Prob. 7E.1DQCh. 7 - Prob. 7E.2DQCh. 7 - Prob. 7E.3DQCh. 7 - Prob. 7E.1AECh. 7 - Prob. 7E.1BECh. 7 - Prob. 7E.2AECh. 7 - Prob. 7E.2BECh. 7 - Prob. 7E.3AECh. 7 - Prob. 7E.3BECh. 7 - Prob. 7E.4AECh. 7 - Prob. 7E.4BECh. 7 - Prob. 7E.5AECh. 7 - Prob. 7E.5BECh. 7 - Prob. 7E.6AECh. 7 - Prob. 7E.6BECh. 7 - Prob. 7E.7AECh. 7 - Prob. 7E.7BECh. 7 - Prob. 7E.8AECh. 7 - Prob. 7E.8BECh. 7 - Prob. 7E.9AECh. 7 - Prob. 7E.9BECh. 7 - Prob. 7E.1PCh. 7 - Prob. 7E.2PCh. 7 - Prob. 7E.3PCh. 7 - Prob. 7E.4PCh. 7 - Prob. 7E.5PCh. 7 - Prob. 7E.6PCh. 7 - Prob. 7E.7PCh. 7 - Prob. 7E.8PCh. 7 - Prob. 7E.9PCh. 7 - Prob. 7E.12PCh. 7 - Prob. 7E.15PCh. 7 - Prob. 7E.16PCh. 7 - Prob. 7E.17PCh. 7 - Prob. 7F.1DQCh. 7 - Prob. 7F.2DQCh. 7 - Prob. 7F.3DQCh. 7 - Prob. 7F.1AECh. 7 - Prob. 7F.1BECh. 7 - Prob. 7F.2AECh. 7 - Prob. 7F.2BECh. 7 - Prob. 7F.3AECh. 7 - Prob. 7F.3BECh. 7 - Prob. 7F.4AECh. 7 - Prob. 7F.4BECh. 7 - Prob. 7F.5AECh. 7 - Prob. 7F.5BECh. 7 - Prob. 7F.6AECh. 7 - Prob. 7F.6BECh. 7 - Prob. 7F.7AECh. 7 - Prob. 7F.7BECh. 7 - Prob. 7F.8AECh. 7 - Prob. 7F.8BECh. 7 - Prob. 7F.9AECh. 7 - Prob. 7F.9BECh. 7 - Prob. 7F.10AECh. 7 - Prob. 7F.10BECh. 7 - Prob. 7F.11AECh. 7 - Prob. 7F.11BECh. 7 - Prob. 7F.12AECh. 7 - Prob. 7F.12BECh. 7 - Prob. 7F.13AECh. 7 - Prob. 7F.13BECh. 7 - Prob. 7F.14AECh. 7 - Prob. 7F.14BECh. 7 - Prob. 7F.1PCh. 7 - Prob. 7F.4PCh. 7 - Prob. 7F.6PCh. 7 - Prob. 7F.7PCh. 7 - Prob. 7F.8PCh. 7 - Prob. 7F.9PCh. 7 - Prob. 7F.10PCh. 7 - Prob. 7F.11PCh. 7 - Prob. 7.3IACh. 7 - Prob. 7.4IACh. 7 - Prob. 7.5IACh. 7 - Prob. 7.6IA
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