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- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_(Nair)/07%3A_Classical_Statistical_Mechanics/7.02%3A_Maxwell-Boltzmann_StatisticsNow we can see how all this applies to the particles in a gas. The analog of heads or tails would be the momenta and other numbers which characterize the particle properties. Thus, we can consider N p...Now we can see how all this applies to the particles in a gas. The analog of heads or tails would be the momenta and other numbers which characterize the particle properties. Thus, we can consider N particles distributed into different cells, each of the cells standing for a collection of observables or quantum numbers which can characterize the particle.
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/12%3A_Temperature_and_Heat/12.05%3A_Heat_Capacity_and_Equipartition_of_EnergyThe only new feature is that you should determine whether the case just presented—ideal gases at constant volume—applies to the problem. (For solid elements, looking up the specific heat capacity is g...The only new feature is that you should determine whether the case just presented—ideal gases at constant volume—applies to the problem. (For solid elements, looking up the specific heat capacity is generally better than estimating it from the Law of Dulong and Petit.) In the case of an ideal gas, determine the number d of degrees of freedom from the number of atoms in the gas molecule and use it to calculate \(C_V\) (or use \(C_V\) to solve for d).
- https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9B__Waves_Sound_Optics_Thermodynamics_and_Fluids/05%3A_Fundamentals_of_Thermodynamics/5.06%3A_Equipartition_of_EnergyWe know that temperature provides a measure of the thermal energy in a system, and that thermal energy is microscopic random mechanical energy. We will see how the temperature change that results fro...We know that temperature provides a measure of the thermal energy in a system, and that thermal energy is microscopic random mechanical energy. We will see how the temperature change that results from an increase of thermal energy relates to the number of "modes" of microscopic mechanical energy are available.
- https://phys.libretexts.org/Courses/Joliet_Junior_College/Physics_201_-_Fall_2019v2/Book%3A_Custom_Physics_textbook_for_JJC/12%3A_Temperature_and_Kinetic_Theory/12.06%3A_The_Kinetic_Theory_of_Gases/Heat_Capacity_and_Equipartition_of_EnergySummary Every degree of freedom of an ideal gas contributes \(\frac{1}{2}k_BT\) per atom or molecule to its changes in internal energy. Every degree of freedom contributes \(\frac{1}{2}R\) to its mol...Summary Every degree of freedom of an ideal gas contributes \(\frac{1}{2}k_BT\) per atom or molecule to its changes in internal energy. Every degree of freedom contributes \(\frac{1}{2}R\) to its molar heat capacity at constant volume \(C_V\) and do not contribute if the temperature is too low to excite the minimum energy dictated by quantum mechanics. Therefore, at ordinary temperatures \(d = 3\) for monatomic gases, \(d = 5\) for diatomic gases, and \(d \approx 6\) for polyatomic gases.
- https://phys.libretexts.org/Courses/Joliet_Junior_College/Physics_201_-_Fall_2019/Book%3A_Physics_(Boundless)/11%3A_Temperature_and_Kinetic_Theory/11.06%3A_The_Kinetic_Theory_of_Gases/Heat_Capacity_and_Equipartition_of_EnergySummary Every degree of freedom of an ideal gas contributes \(\frac{1}{2}k_BT\) per atom or molecule to its changes in internal energy. Every degree of freedom contributes \(\frac{1}{2}R\) to its mol...Summary Every degree of freedom of an ideal gas contributes \(\frac{1}{2}k_BT\) per atom or molecule to its changes in internal energy. Every degree of freedom contributes \(\frac{1}{2}R\) to its molar heat capacity at constant volume \(C_V\) and do not contribute if the temperature is too low to excite the minimum energy dictated by quantum mechanics. Therefore, at ordinary temperatures \(d = 3\) for monatomic gases, \(d = 5\) for diatomic gases, and \(d \approx 6\) for polyatomic gases.
- https://phys.libretexts.org/Courses/University_of_California_Davis/Physics_9B_Fall_2020_Taufour/05%3A_Fundamentals_of_Thermodynamics/5.06%3A_Equipartition_of_EnergyWe know that temperature provides a measure of the thermal energy in a system, and that thermal energy is microscopic random mechanical energy. We will see how the temperature change that results fro...We know that temperature provides a measure of the thermal energy in a system, and that thermal energy is microscopic random mechanical energy. We will see how the temperature change that results from an increase of thermal energy relates to the number of "modes" of microscopic mechanical energy are available.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04%3A_Statistical_Ensembles/4.07%3A_Ideal_Gas_Statistical_Mechanics\[\begin{split} Z(T,V,N)&={1\over N!}\prod_{i=1}^N\int\!{d^d\!x\ns_i\, d^d\!p\ns_i\over (2\pi\hbar)^d}\>e^{-\beta\Bp_i^2/2m}\\ &={V^N\over N!}\left(\int\limits_{-\infty}^\infty\!\!{dp\over 2\pi\hbar}\...\[\begin{split} Z(T,V,N)&={1\over N!}\prod_{i=1}^N\int\!{d^d\!x\ns_i\, d^d\!p\ns_i\over (2\pi\hbar)^d}\>e^{-\beta\Bp_i^2/2m}\\ &={V^N\over N!}\left(\int\limits_{-\infty}^\infty\!\!{dp\over 2\pi\hbar}\>e^{-\beta p^2/2m}\right)^{\!\!Nd}\\ &={1\over N!}\bigg({V\over\lambda_T^d}\bigg)^{\!N}\ , \end{split}\]
- https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/02%3A_The_Kinetic_Theory_of_Gases/2.04%3A_Heat_Capacity_and_Equipartition_of_EnergySummary Every degree of freedom of an ideal gas contributes \(\frac{1}{2}k_BT\) per atom or molecule to its changes in internal energy. Every degree of freedom contributes \(\frac{1}{2}R\) to its mol...Summary Every degree of freedom of an ideal gas contributes \(\frac{1}{2}k_BT\) per atom or molecule to its changes in internal energy. Every degree of freedom contributes \(\frac{1}{2}R\) to its molar heat capacity at constant volume \(C_V\) and do not contribute if the temperature is too low to excite the minimum energy dictated by quantum mechanics. Therefore, at ordinary temperatures \(d = 3\) for monatomic gases, \(d = 5\) for diatomic gases, and \(d \approx 6\) for polyatomic gases.
