CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A...

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CHM 421/621 Statistical Mechanics Lecture 3

Transcript of CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A...

Page 1: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities.

CHM 421/621 Statistical Mechanics

Lecture 3

Page 2: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities.

Introduction and ReviewLecture Plan

An example application of Statistical Mechanics

Page 3: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities.

Need for Statistical Mechanics

How do we bring in the microscopic details into consideration?

Basic Assumption:

The physical properties of a macroscopic system depend only on the average behaviour of all the atoms in that system.

Program of Statistical Mechanics

(1) Assume a microscopic model for the system.(2) Assume/Derive a probability distribution for the various micro states possible

under the given external constraints (thus bringing in the macroscopic variables).(3) Compute weighted averages of desired properties.

Page 4: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities.

A familiar exampleClassical microstates - set of instantaneous coordinates and velocities.

System: Consider a system of N particles moving independently of each other and inside a closed container of volume V. The container (and hence the particles) is assumed to be in thermal contact with a reservoir at temperature T.

Model:

Probability distribution:

Momentum of ith particlePosition of ith particle

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� =1

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H({ri(t),pi(t)}) =NX

i=1

1

2mp2i

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⇢({ri,pi}) =1

Q(N,V, T )exp (��H({ri,pi}))

<latexit sha1_base64="TMTWjhkuQZ5ZKkvyPVF46k+fusM=">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</latexit><latexit sha1_base64="TMTWjhkuQZ5ZKkvyPVF46k+fusM=">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</latexit><latexit sha1_base64="TMTWjhkuQZ5ZKkvyPVF46k+fusM=">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</latexit><latexit sha1_base64="TMTWjhkuQZ5ZKkvyPVF46k+fusM=">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</latexit>

Page 5: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities.

A familiar exampleProbability distribution:

Meaning: The probability of finding the system in a microstate with the momenta of the particles between pi and pi + dpi and their positions between ri and ri + dri is given by

=>

So what is Q(N,V,T)?

⇢({ri,pi}) =1

Q(N,V, T )exp (��H({ri,pi}))

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⇢ ({ri,pi}) d3Nrid3Npi

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3Npi = 1<latexit sha1_base64="v1/5oNEvw6EHMFyzCdaOcql8Vdc=">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</latexit><latexit sha1_base64="v1/5oNEvw6EHMFyzCdaOcql8Vdc=">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</latexit><latexit sha1_base64="v1/5oNEvw6EHMFyzCdaOcql8Vdc=">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</latexit><latexit sha1_base64="v1/5oNEvw6EHMFyzCdaOcql8Vdc=">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</latexit>

Page 6: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities.

A familiar exampleQ(N,V, T ) = AV N

✓2⇡m

�

◆ 3N2

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Here A is a constant to ensure the correct dimensions. We will show later that

In this case, the normalisation equation becomes

Q(N,V, T ) =

✓V

⇤3

◆N

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With

A =1

h3N<latexit sha1_base64="8PXXfFi46nz9vdCR3ow7deh7HhU=">AAAB/XicbVDLSsNAFL3xWesrPnZuBovgqiQq6EaounElFewD2lgm00k7dDIJMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPHnCntON/W3PzC4tJyYaW4ura+sWlvbddVlEhCayTikWz6WFHOBK1ppjltxpLi0Oe04Q+uRn7jgUrFInGnhzH1QtwTLGAEayN17N0LdI7agcQkdbO0f58e32RZxy45ZWcMNEvcnJQgR7Vjf7W7EUlCKjThWKmW68TaS7HUjHCaFduJojEmA9yjLUMFDqny0vH1GTowShcFkTQlNBqrvydSHCo1DH3TGWLdV9PeSPzPayU6OPNSJuJEU0Emi4KEIx2hURSoyyQlmg8NwUQycysifWyi0CawognBnX55ltSPyq5Tdm9PSpXLPI4C7ME+HIILp1CBa6hCDQg8wjO8wpv1ZL1Y79bHpHXOymd24A+szx9hdJSJ</latexit><latexit sha1_base64="8PXXfFi46nz9vdCR3ow7deh7HhU=">AAAB/XicbVDLSsNAFL3xWesrPnZuBovgqiQq6EaounElFewD2lgm00k7dDIJMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPHnCntON/W3PzC4tJyYaW4ura+sWlvbddVlEhCayTikWz6WFHOBK1ppjltxpLi0Oe04Q+uRn7jgUrFInGnhzH1QtwTLGAEayN17N0LdI7agcQkdbO0f58e32RZxy45ZWcMNEvcnJQgR7Vjf7W7EUlCKjThWKmW68TaS7HUjHCaFduJojEmA9yjLUMFDqny0vH1GTowShcFkTQlNBqrvydSHCo1DH3TGWLdV9PeSPzPayU6OPNSJuJEU0Emi4KEIx2hURSoyyQlmg8NwUQycysifWyi0CawognBnX55ltSPyq5Tdm9PSpXLPI4C7ME+HIILp1CBa6hCDQg8wjO8wpv1ZL1Y79bHpHXOymd24A+szx9hdJSJ</latexit><latexit sha1_base64="8PXXfFi46nz9vdCR3ow7deh7HhU=">AAAB/XicbVDLSsNAFL3xWesrPnZuBovgqiQq6EaounElFewD2lgm00k7dDIJMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPHnCntON/W3PzC4tJyYaW4ura+sWlvbddVlEhCayTikWz6WFHOBK1ppjltxpLi0Oe04Q+uRn7jgUrFInGnhzH1QtwTLGAEayN17N0LdI7agcQkdbO0f58e32RZxy45ZWcMNEvcnJQgR7Vjf7W7EUlCKjThWKmW68TaS7HUjHCaFduJojEmA9yjLUMFDqny0vH1GTowShcFkTQlNBqrvydSHCo1DH3TGWLdV9PeSPzPayU6OPNSJuJEU0Emi4KEIx2hURSoyyQlmg8NwUQycysifWyi0CawognBnX55ltSPyq5Tdm9PSpXLPI4C7ME+HIILp1CBa6hCDQg8wjO8wpv1ZL1Y79bHpHXOymd24A+szx9hdJSJ</latexit><latexit sha1_base64="8PXXfFi46nz9vdCR3ow7deh7HhU=">AAAB/XicbVDLSsNAFL3xWesrPnZuBovgqiQq6EaounElFewD2lgm00k7dDIJMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPHnCntON/W3PzC4tJyYaW4ura+sWlvbddVlEhCayTikWz6WFHOBK1ppjltxpLi0Oe04Q+uRn7jgUrFInGnhzH1QtwTLGAEayN17N0LdI7agcQkdbO0f58e32RZxy45ZWcMNEvcnJQgR7Vjf7W7EUlCKjThWKmW68TaS7HUjHCaFduJojEmA9yjLUMFDqny0vH1GTowShcFkTQlNBqrvydSHCo1DH3TGWLdV9PeSPzPayU6OPNSJuJEU0Emi4KEIx2hURSoyyQlmg8NwUQycysifWyi0CawognBnX55ltSPyq5Tdm9PSpXLPI4C7ME+HIILp1CBa6hCDQg8wjO8wpv1ZL1Y79bHpHXOymd24A+szx9hdJSJ</latexit>

⇤ =hp

2⇡mkBT<latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit>

1

h3N

Z⇢ ({ri,pi}) d3Nrid

3Npi = 1<latexit sha1_base64="Nv6asxcmd7vZVKasPF6uZLXmHew=">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</latexit><latexit sha1_base64="Nv6asxcmd7vZVKasPF6uZLXmHew=">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</latexit><latexit sha1_base64="Nv6asxcmd7vZVKasPF6uZLXmHew=">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</latexit><latexit sha1_base64="Nv6asxcmd7vZVKasPF6uZLXmHew=">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</latexit>

Page 7: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec3.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities.

Average Energy

U = hHi =Z

dr

Zdp H({ri,pi}) ⇢({ri,pi})

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U =3

2NkBT

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Calculate

Answer


CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/Courses/CHM421/Lectures/Lec22.pdfLecture 22 Formalisms of Statistical Mechanics Lecture Plan Basic Postulates Counting microstates

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CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/Courses/CHM421/Lectures/Lec7.pdf-1 -0.5 0 0.5 1 x 0 1 2 3 4 f(x) alpha = 20 alpha = 40 Thus, the smaller the value of the fatter

CHM 421 HETEROCYCLIC CHEMISTRY - Home | …nouedu.net/sites/default/files/2017-03/CHM 421...Introduction CHM421: Heterocyclic Chemistry is a 2 -credit course for B.Sc. Chemistry. The

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CHM 421 HETEROCYCLIC CHEMISTRY - nou.edu.ng · CHM421: Heterocyclic Chemistry is a 2 -credit course for B.Sc. Chemistry. The course is broken down into 3 modules of 15 study units.

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CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities.

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CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec5.pdfNature of the distribution hOi = 1 Q Z dr Z dp O(r, p)exp(H (r, p))

CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec5.pdfNature of the distribution hOi = 1 Q Z dr Z dp O(r, p)exp(H (r, p))

It makes all of us proud to say that your SAFE HANDS contributed 11/60 results in MGIMS Vardha through an exam. We have 42 + students with more than 500.

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It makes all of us proud to say that your SAFE HANDS contributed 13/60 results in MGIMS Vardha through exam. Have 47 students out of 122 with more than.

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CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/Courses/CHM421/Lectures/Lec31.pdfz exp ~2k2 2m = V 3 0 @ Z1 0 dk x exp ~2k2 x 2m 1 A 3 = V 8 3 2 mk B T ~2 3 2 = V 3

CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/Courses/CHM421/Lectures/Lec31.pdfz exp ~2k2 2m = V 3 0 @ Z1 0 dk x exp ~2k2 x 2m 1 A 3 = V 8 3 2 mk B T ~2 3 2 = V 3

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