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Title+Summary	Name	Date
Problem-TH-04002 Node id: 2246page The molar energy of a monoatomic gas which obeys van der Waal's equation is given by \( E= \frac{3}{2}kT - \frac{a}{v}\), where \(V\) is the volume at temperature \(T\), and \(a\) is a constant. Initially one mole of gas is at temperature \(T_1\) and occupies volume \(V_1\). The gas is allowed to expand adiabatically into a vacuum so that it occpies a total volume \(V_2\). What is the final temperature of the gas? MANDL	kapoor	20-02-08 16:02:01	n
CM-Mod09 Rigid BodyDynamics Node id: 2206page	kapoor	20-02-08 16:02:01	n
Problem-TH-03001 Work done; van der Waals gas Node id: 2200page Consider \(N\) molecules of a gas obeying van der Waals equation of state given by \[\left(P+ \frac{a N^2}{V^2}\right)\big(V-Nb\big) = Nk_B T\] where \(a\) is a measure of the attractive forces between the molecules and \(b\) is another constant proportional to the size of a molecule. The other symbols have their usual meanings. Show that during an isothermal expansion (Temperature is kept onstant) from volume \(V_1\) to volume \(V_2\) quasi-statically and reversibly, the work done is \[ W =-Nk_B T \log\left(\frac{V_2-Nb}{V_1-Nb}\right) + a^2 \Big(\frac{1}{V_1}-\frac{1}{V_2}\Big)\]	kapoor	20-02-08 16:02:01	n
Problem-TH-04001 Fiirst Law Relation between \(C_p, C_v\). Node id: 2199page Starting from the first law of thermodynamics show that \[C_p - C_v = \Big[P +\Big(\frac{\partial U}{\partial V}\Big)_T\Big] \Big(\frac{\partial V}{\partial T}\Big)_P\] In the above \(C_p\) : heat capacity at constant pressure; \(C_v\) : heat capacity at constant volume; \(U\) : internal energy: \(V\) is the volume. For an ideal gas show that the above reduces to \(C_p -C_v = Nk_B\) , where \(N\) is the number of molecules and \(k_B\) is the Boltzmann constant.	kapoor	20-02-08 16:02:01	n
QM-02-04 Vetor Spaces Node id: 2185page	kapoor	20-02-08 16:02:01	n
CM-Mod03 :: Action Principle Node id: 2015page	kapoor	20-02-08 16:02:01	n
QM-Mod06 :: General Principles of Quantum Mechanics Node id: 1975page Repository of problems on Postulates of Quantum Mechanics. All problems fall under "Analysis and Application" levels of Bloom's Taxonomy. Use navigation links at the right bottom of the page	kapoor	20-02-08 16:02:01	n
Question-QM-06003 --- Computing probability Node id: 1966page	kapoor	20-02-08 16:02:01	n
Question-QM-06001 ---- Commuting Set of Operators Node id: 1964page	kapoor	20-02-08 16:02:01	n
CM-Mod02 :: Euler Lagrange Equations of Motion Node id: 1958page	kapoor	20-02-08 16:02:01	n
CM-Mod01:: Selected Problems in Newtonian Mechanics Node id: 1957page Bloom Category :Application and Anaysis The problems relating to bounded motion in one dimension appear here.	kapoor	20-02-08 16:02:01	n
CM-Mod10 Canonical Transformations Node id: 1955page	kapoor	20-02-08 16:02:01	n
QM-01 :: Review of Classical Theories Node id: 1380page	kapoor	20-02-08 16:02:01	n
\(\S\S\) 7.15 Q[2] Node id: 2830page	kapoor	20-02-08 16:02:37	n
\(\S\S\) 7.15 Q[1] Node id: 2829page	kapoor	20-02-08 16:02:37	n
Complex Variables --- Principles and Problem Sessions === SOLUTIONS and ERRATA Node id: 2193curated_content It is proposed to post solutions to almost all the problem session in the book here. At present solutions to all most all Problem Sessions of Chapter 7, Contour Integration, has been completed. The solutions to problems in other chapters is being taken up. At present solutions to problems in Chapter 3. "Function with Branch Point Singularity" is Being Uploaded. Errata is being compiled and will be uploaded in this book hierarchy. Click to see what is available. I take this opportunity to thank my friend and colleague, Prof. T. Amarnath, from School of Mathematics, University of Hyderabad for his kind words of appreciation and for recommending the book to National Board of Higher Mathematics for inclusion in scheme of distribution of mathematics books to Universities and Institutes in India.	kapoor	20-02-08 16:02:37	n
\(\S\S\) 7.6 Q[2] \(\int_0^\infty \frac{x^{p-1}}{x^2+2x+2}\) Node id: 1863page	kapoor	20-02-08 16:02:37	n
\(\S\S\) 7.4 Q[9] \( \int_0^\infty \frac{\cos a x \, dx}{(x^2+b^2)^2+c^2}\) Node id: 1857page	kapoor	20-02-08 16:02:37	n
\(\S\S\) 7.3 Q[1] \(\int_0^\infty\frac{1}{(x^2+p^2)^2} dx \) Node id: 1566page Full details are written out for this problem. This includes a proof that certain integrals vanish when Darboux theorem is used. In solutions to all other similar problems of this section, some of these details, being repetitive in nature, are suppressed. It is hoped that, if required, the reader will be able to supply the details by consulting this solution with full details.	kapoor	20-02-08 16:02:37	n
Pages under construction Node id: 3447page	kapoor	20-02-08 16:02:30	n

The molar energy of a monoatomic gas which obeys van der Waal's equation is given by
\( E= \frac{3}{2}kT - \frac{a}{v}\),
where \(V\) is the volume at temperature \(T\), and \(a\) is a constant. Initially one mole of gas is at temperature \(T_1\) and occupies volume \(V_1\). The gas is allowed to expand adiabatically into a vacuum so that it occpies a total volume \(V_2\). What is the final temperature of the gas?

MANDL

Consider \(N\) molecules of a gas obeying van der Waals equation of state given by
\[\left(P+ \frac{a N^2}{V^2}\right)\big(V-Nb\big) = Nk_B T\]
where \(a\) is a measure of the attractive forces between the molecules and \(b\) is another constant proportional to the size of a molecule. The other symbols have their usual meanings. Show that during an isothermal expansion (Temperature is kept onstant) from volume \(V_1\) to volume \(V_2\) quasi-statically and reversibly, the work done is
\[ W =-Nk_B T \log\left(\frac{V_2-Nb}{V_1-Nb}\right) + a^2 \Big(\frac{1}{V_1}-\frac{1}{V_2}\Big)\]

Starting from the first law of thermodynamics show that
\[C_p - C_v = \Big[P +\Big(\frac{\partial U}{\partial V}\Big)_T\Big]
\Big(\frac{\partial V}{\partial T}\Big)_P\]
In the above \(C_p\) : heat capacity at constant pressure; \(C_v\) :
heat capacity at constant volume; \(U\) : internal energy: \(V\) is
the volume. For an ideal gas show that the above reduces
to \(C_p -C_v = Nk_B\) , where \(N\) is the number of molecules and
\(k_B\) is the Boltzmann constant.

Repository of problems on Postulates of Quantum Mechanics.
All problems fall under "Analysis and Application" levels
of Bloom's Taxonomy.

Use navigation links at the right bottom of the page

Bloom Category :Application and Anaysis

The problems relating to bounded motion in one dimension appear here.

It is proposed to post solutions to almost all the problem session in the book here. At present solutions to all most all Problem Sessions of Chapter 7, Contour Integration, has been completed. The solutions to problems in other chapters is being taken up.
At present solutions to problems in Chapter 3. "Function with Branch Point Singularity"
is Being Uploaded.

Errata is being compiled and will be uploaded in this book hierarchy.
Click to see what is available.

I take this opportunity to thank my friend and colleague, Prof. T. Amarnath, from School of Mathematics, University of Hyderabad for his kind words of appreciation and for recommending the book to National Board of Higher Mathematics for inclusion in scheme of distribution of mathematics books to Universities and Institutes in India.

Full details are written out for this problem. This includes a proof that certain integrals vanish when Darboux theorem is used. In solutions to all other similar problems of this section, some of these details, being repetitive in nature, are suppressed. It is hoped that, if required, the reader will be able to supply the details by consulting this solution with full details.

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Problem-TH-04002

CM-Mod09 Rigid BodyDynamics

Problem-TH-03001 Work done; van der Waals gas

Problem-TH-04001 Fiirst Law Relation between \(C_p, C_v\).

QM-02-04 Vetor Spaces

CM-Mod03 :: Action Principle

QM-Mod06 :: General Principles of Quantum Mechanics

Question-QM-06003 --- Computing probability

Question-QM-06001 ---- Commuting Set of Operators

CM-Mod02 :: Euler Lagrange Equations of Motion

CM-Mod01:: Selected Problems in Newtonian Mechanics

CM-Mod10 Canonical Transformations

QM-01 :: Review of Classical Theories

\(\S\S\) 7.15 Q[2]

\(\S\S\) 7.15 Q[1]

Complex Variables --- Principles and Problem Sessions === SOLUTIONS and ERRATA

\(\S\S\) 7.6 Q[2] \(\int_0^\infty \frac{x^{p-1}}{x^2+2x+2}\)

\(\S\S\) 7.4 Q[9] \( \int_0^\infty \frac{\cos a x \, dx}{(x^2+b^2)^2+c^2}\)

\(\S\S\) 7.3 Q[1] \(\int_0^\infty\frac{1}{(x^2+p^2)^2} dx \)

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