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[NOTES/SM-04007] Applications of Maxwell's DistributionThe equation of state of a perfect gas is derived using Maxwell distribution of velocities of molecules in a perfect gas. The Maxwell distribution can be verified by means of an experiment on effusion of a gas through a hole. This also lead to determination of the Boltzmann constant |
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24-04-04 12:04:48 |
[NOTES/SM-04006] Maxwell Distribution of speeds in an ideal gasFor an ideal gas Maxwell's distribution of velocities is obtained using canonical ensemble. |
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24-04-04 12:04:24 |
[NOTES/SM-04012] Statistical EntropyIt is shown that the statistical entropy coincides with the thermodynamic expression for entropy. |
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24-04-04 11:04:16 |
[NOTES/SM-04004] Thermodynamic Functions in Terms of Canonical Partition FunctionUsing the expression for energy in terms of the canonical partition function and the \(TdS\) equation \[dU=TdS -pdV\] we obtain an expression for the entropy,.<$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} $ |
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24-04-04 10:04:24 |
[NOTES/SM-04001] The Canonical EnsembleIn this lecture canonical ensemble and canonical partition function are introduced. This topic has a central place in equilibrium statistical mechanics. This ensemble describes the microstates of a system in equilibrium with a heat bath at temperature \(T\). The probability of a microstate having energy \(E\) is proportional to \(\exp(-\beta E\), where \(\beta=kT\) and \(k\) is Boltzmann constant. |
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24-04-04 10:04:00 |
[NOTES/SM-04017] Internal Energy$\newcommand{\pp}[2][]{\frac {\partial #1}{\partial #2}}$For a system in contact with heat bath, the energy is not well defined. The system being in a micro state with energy value \(E_k\) is given by Boltzmann distribution. We compute average energy and show that the average energy it in terms of the canonical partition function is given by \[ U = -k \pp[\ln Z]{\beta}.\] It is shown that for macroscopic systems the variance of energy is negligible compared to the energy, but not for microscopic systems. Hence \(\bar E\) can be identified with the thermodynamic internal energy. |
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24-04-04 06:04:35 |
[NOTES/SM-03004] Application of Micro Canonical Ensemble to Ideal GasAs an application of micro canonical ensemble, the ideal gas equation and law of equipartition of energy are derived. $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ |
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24-04-02 06:04:14 |
[NOTES/SM-03003] Equilibrium ConditionsUsing the principle of maximum entropy, we derive conditions so that an isolated system consisting of two systems may be in equilibrium. |
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24-04-02 06:04:49 |
[NOTES/SM-03002] Application of Micro canonical Ensemble to Two Level SystemAs an application of micro canonical ensemble we show that the fraction of particles in the second level at temperature \(T\) is given by \begin{equation} \frac{n}{N} = \frac{1}{e^{\epsilon/kT} +1}. \end{equation} $newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ |
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24-03-31 14:03:40 |
[NOTES/SM-03001] Fundamental Postulate of Statistical MechanicsThe postulate of equal a priori probability, the Boltzmann equation and the principle of increase of entropy and their role in statistical mechanics are briefly explained. |
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24-03-31 14:03:54 |