MohammedRady97 said:While I am familiar with the derivation of the relationship between mean translational kinetic energy, ##<E_k>##, and temperature, ##T##, for an ideal gas (namely ##<E_k> = \frac{3}{2}k_B T##); I fail to see how we can establish such a relationship for solids, liquids, and even real gases. Could someone please guide me through this?
The concept of temperature is tied to heat flow. If heat flows spontaneously from body A to body B we say that body B has a lower temperature than body A. The heat flow stops when the molecules in both bodies reach a common thermal equilibrium. At the molecular level, the heat flow from one body to another (ie. the transfer of molecular kinetic energy) results from a difference in the translational kinetic energies of the molecules (i.e. molecular collisions). If the molecules were just vibrating or rotating about their centres of mass there would be no flow of kinetic energy from one molecule to another.MohammedRady97 said:While I am familiar with the derivation of the relationship between mean translational kinetic energy, ##<E_k>##, and temperature, ##T##, for an ideal gas (namely ##<E_k> = \frac{3}{2}k_B T##); I fail to see how we can establish such a relationship for solids, liquids, and even real gases. Could someone please guide me through this?
Andrew Mason said:When the bodies reach thermal equilibrium, the total kinetic energies of the molecules will be equally partitioned between all modes (ie. active vibrational, active rotational and translational motion). But the transfer of energy is due to the differences in translational kinetic energies. This applies whether you are dealing with ideal gases, real gases, solids or liquids.
AM
Vibhor said:How is it that first thermal equilibrium is reached and then transferred energy is partitioned in different modes ?
As heat flow occurs from body A to body B (due to higher average translational kinetic energy of the molecules in body A compared to body B) the energy in body A decreases and that of body B increases. When thermal equilibrium is reached the internal energy in each of body A and B is equally partitioned between all modes. Before thermal equilibrium is reached, all modes will have energy but they will not necessarily be equally partitioned.Vibhor said:Hi Andrew ,
Could you please explain the first line , more specifically "When the bodies reach thermal equilibrium" ?
My understanding is that " the energy transferred would be equally partitioned between all modes until the average translational kinetic energy of the two bodies become equal . As a consequence of which the bodies reach thermal equilibrium ". But you have written that after the bodies reach thermal equilibrium ,then energy is partitioned .
How is it that first thermal equilibrium is reached and then t ransferred energy is partitioned in different modes ?
Temperature is directly proportional to mean translational kinetic energy. This means that as temperature increases, the average speed of molecules also increases, resulting in a higher mean translational kinetic energy. Conversely, as temperature decreases, the average speed of molecules decreases, resulting in a lower mean translational kinetic energy.
The relationship between temperature and mean translational kinetic energy is described by the kinetic theory of gases. This theory states that the average kinetic energy of gas molecules is directly proportional to the temperature of the gas. Therefore, as temperature increases, the mean translational kinetic energy also increases.
The mass of molecules does not affect the relationship between temperature and mean translational kinetic energy. This is because the kinetic energy of a molecule is determined by its temperature, not its mass. However, heavier molecules may have a lower average speed at the same temperature compared to lighter molecules, resulting in a lower mean translational kinetic energy.
No, temperature and mean translational kinetic energy cannot be used interchangeably. Temperature measures the average kinetic energy of a system, while mean translational kinetic energy specifically measures the average kinetic energy of the translational motion of molecules. They are related, but they are not the same measurement.
When the mean translational kinetic energy of a gas increases, its temperature also increases. This is because the average speed of molecules is directly proportional to both temperature and mean translational kinetic energy. Therefore, an increase in mean translational kinetic energy results in an increase in temperature.