It is convenient to consider a particular physical phenomenon or class of phenomena using models of varying degrees of approximation. For example, when describing the behavior of a gas, a physical model is used - an ideal gas.
Any model has limits of applicability, beyond which its refinement or the use of more complex options is required. Here we consider a simple case of describing the internal energy of a physical system based on the most significant properties of gases within certain limits.
Perfect gas
For the convenience of describing some fundamental processes, this physical model simplifies the real gas as follows:
- Neglects the size of gas molecules. This means that there are phenomena for which an adequate description of this parameter is not essential.
- He neglects intermolecular interactions, that is, accepts that in the processes of interest to them they appear at negligible intervals and do not affect the state of the system. In this case, the interactions are in the form of an absolutely elastic shock, in which there is no energy loss on deformation.
- Neglects the interaction of molecules with the walls of the tank.
- Assumes that the gas-reservoir system is characterized by thermodynamic equilibrium.
Such a model is suitable for describing real gases if the pressures and temperatures are relatively low.
Energy state of the physical system
Every macroscopic physical system (body, gas or liquid in a vessel) has, in addition to its own kinetic and potential, another type of energy - internal. This value is obtained by summing the energies of all the subsystems of the physical system - molecules.
Each molecule in the gas also has its own potential and kinetic energy. The latter is due to the continuous chaotic thermal motion of the molecules. Various interactions between them (electrical attraction, repulsion) are determined by potential energy.
It must be remembered that if the energy state of any part of the physical system does not have any effect on the macroscopic state of the system, then it is not taken into account. For example, under ordinary conditions, nuclear energy does not manifest itself in changes in the state of a physical object, so it does not need to be taken into account. But at high temperatures and pressures, this is already necessary.
Thus, the internal energy of the body reflects the nature of the movement and interaction of its particles. This means that this term is synonymous with the commonly used concept of "thermal energy".
Monoatomic ideal gas
Monatomic gases, that is, those whose atoms are not combined into molecules, exist in nature - these are inert gases. Gases, such as oxygen, nitrogen, or hydrogen, can exist in such a state only under conditions when energy is expended from the outside to constantly restore this state, since their atoms are chemically active and tend to combine into a molecule.
Consider the energy state of a monatomic ideal gas placed in a vessel of a certain volume. This is the simplest case. We remember that the electromagnetic interaction of atoms with each other and with the walls of the vessel, and, consequently, their potential energy is negligible. So the internal energy of a gas includes only the sum of the kinetic energies of its atoms.
It can be calculated by multiplying the average kinetic energy of atoms in a gas by their number. The average energy is E = 3/2 x R / N A x T, where R is the universal gas constant, N A is the Avogadro number, T is the absolute temperature of the gas. The number of atoms is calculated by multiplying the amount of substance by the Avogadro constant. The internal energy of a monatomic gas will be equal to U = N A x m / M x 3/2 x R / N A x T = 3/2 x m / M x RT. Here m is the mass and M is the molar mass of the gas.
Assume that the chemical composition of the gas and its mass always remain the same. In this case, as can be seen from our formula, the internal energy depends only on the gas temperature. For a real gas, it will be necessary to take into account, in addition to temperature, a change in volume, since it affects the potential energy of atoms.
Molecular gases
In the above formula, the number 3 characterizes the number of degrees of freedom of motion of a monatomic particle - it is determined by the number of coordinates in space: x, y, z. For a state of a monatomic gas, it does not matter whether its atoms rotate.
Molecules are spherically asymmetric, therefore, when determining the energy state of molecular gases, the kinetic energy of their rotation must be taken into account. Diatomic molecules, in addition to the listed degrees of freedom associated with translational motion, have two more associated with rotation around two mutually perpendicular axes; polyatomic molecules have three such independent axes of rotation. Therefore, particles of diatomic gases are characterized by the number of degrees of freedom f = 5, while in polyatomic molecules f = 6.
Due to the randomness inherent in thermal motion, all directions of both rotational and translational motion are completely equally probable. The average kinetic energy introduced by each type of motion is the same. Therefore, we can substitute the value f into the formula, which allows us to calculate the internal energy of an ideal gas of any molecular composition: U = f / 2 x m / M x RT.
Of course, we see from the formula that this value depends on the amount of substance, that is, on how much and what kind of gas we took, as well as on the structure of the molecules of this gas. However, since we agreed not to change the mass and chemical composition, we only need to take into account the temperature.
Now consider how the value of U is related to other characteristics of the gas - volume and pressure.
Internal energy and thermodynamic state
Temperature, as you know, is one of the parameters of the thermodynamic state of the system (in this case, gas). In an ideal gas, it is related to pressure and volume by the ratio PV = m / M x RT (the so-called Clapeyron-Mendeleev equation). Temperature determines thermal energy. So the latter can be expressed through a set of other state parameters. She is indifferent to the previous state, as well as to the way it is changed.
Let's see how the internal energy changes when the system goes from one thermodynamic state to another. Its change at any such transition is determined by the difference between the initial and final values. If the system after some intermediate state returned to the original, then this difference will be equal to zero.
Suppose we heated a gas in a tank (that is, added additional energy to it). The thermodynamic state of the gas has changed: its temperature and pressure have increased. Such a process goes without changing the volume. The internal energy of our gas has increased. After that, our gas gave up the supplied energy, cooling to its original state. A factor such as, for example, the speed of these processes will not make any difference. The resulting change in the internal energy of the gas at any heating and cooling rate is zero.
An important point is that the same value of thermal energy may correspond not to one but several thermodynamic states.
The nature of the change in thermal energy
In order to change energy, you need to do work. Work can be done by the gas itself or by external force.
In the first case, the expenditure of energy to complete the work is made due to the internal energy of the gas. For example, we had compressed gas in a tank with a piston. If you let go of the piston, the expanding gas will lift it, doing the job (so that it is useful, let the piston lift some kind of load). The internal energy of the gas will decrease by the amount spent on working against gravity and friction: U 2 = U 1 - A. In this case, the gas work is positive, since the direction of the force applied to the piston coincides with the direction of movement of the piston.
We begin to lower the piston, doing work against the gas pressure force and again against the friction forces. Thus, we will give the gas a certain amount of energy. Here, the work of external forces is already considered positive.
In addition to mechanical work, there is such a way to take away from the gas or give it energy, such as heat transfer (heat transfer). We have already met him in the example of heating gas. The energy transferred to the gas during heat transfer processes is called the amount of heat. Heat transfer is of three types: thermal conductivity, convection and radiant transfer. Let's consider them a little more in detail.
Thermal conductivity
The ability of a substance to heat exchange, carried out by its particles by transferring kinetic energy to each other during mutual collisions during thermal motion, is thermal conductivity. If a certain region of a substance is heated, that is, it is given a certain amount of heat, the internal energy will be distributed homogeneously between all particles on average through collisions of atoms or molecules.
It is clear that thermal conductivity strongly depends on the frequency of collisions, and that, in turn, depends on the average distance between particles. Therefore, gas, especially ideal, is characterized by very low thermal conductivity, and this property is often used for thermal insulation.
Of the real gases, the thermal conductivity is higher for those whose molecules are the lightest and at the same time polyatomic. Molecular hydrogen is the most suitable for this condition, and radon is the least, as the heaviest monatomic gas. The more rarefied the gas, the worse heat conductor it is.
In general, energy transfer due to thermal conductivity for an ideal gas is a very inefficient process.
Convection
Much more efficient for a gas is this type of heat transfer, such as convection, in which the internal energy is distributed through the flow of matter circulating in the gravitational field. The upward flow of hot gas is formed due to the Archimedean force, since it is less dense due to thermal expansion. Moving upward hot gas is constantly being replaced by colder gas - the circulation of gas flows is established. Therefore, in order to ensure efficient, that is, the fastest, heating through convection, it is necessary to heat the gas tank from below - as well as the kettle with water.
If it is necessary to take away a certain amount of heat from the gas, then it is more efficient to place the refrigerator at the top, since the gas that gave energy to the refrigerator will rush down under the influence of gravity.
An example of convection in gas is the heating of indoor air using heating systems (they are placed in the room as low as possible) or cooling using an air conditioner, and under natural conditions, the phenomenon of thermal convection causes the movement of air masses and affects the weather and climate.
In the absence of gravity (in zero gravity in a spaceship), convection, that is, the circulation of air currents, is not established. So it makes no sense to light gas burners or matches on board the spacecraft: hot combustion products will not be taken up, and oxygen will not be brought to the source of the fire, and the flame will die out.
Radiant carry
A substance can also heat up under the influence of thermal radiation, when atoms and molecules acquire energy by absorbing electromagnetic quanta - photons. At low photon frequencies, this process is not very effective. Recall that when we open the microwave, we find there hot foods, but not hot air. With an increase in the radiation frequency, the effect of radiant heating increases, for example, in the upper atmosphere of the Earth, a very rarefied gas is intensively heated and ionized by solar ultraviolet light.
Different gases absorb thermal radiation to varying degrees. So, water, methane, carbon dioxide absorb it quite strongly. The phenomenon of the greenhouse effect is based on this property.
The first law of thermodynamics
Generally speaking, a change in internal energy through heating a gas (heat transfer) also comes down to the work of either gas molecules or above them by an external force (which is also denoted by the opposite sign). What kind of work is done with this method of transition from one state to another? The law of conservation of energy will help us answer this question, more precisely, its concretization in relation to the behavior of thermodynamic systems is the first principle of thermodynamics.
The law, or the universal principle of conservation of energy, in the most generalized form states that energy does not arise from nothing and does not disappear without a trace, but only passes from one form to another. With regard to a thermodynamic system, this should be understood so that the work performed by the system is expressed through the difference between the amount of heat communicated to the system (ideal gas) and the change in its internal energy. In other words, the amount of heat communicated to the gas is expended on this change and on the operation of the system.
In the form of formulas, this is written much easier: dA = dQ - dU, and accordingly, dQ = dU + dA.
We already know that these quantities do not depend on the way in which the transition between states occurs. The speed of this transition and, as a consequence, the efficiency depends on the method.
As for the second law of thermodynamics, it sets the direction of change: heat cannot be transferred from a colder (and therefore less energetic) gas to hotter without additional energy costs from the outside. The second beginning also indicates that part of the energy consumed by the system to complete the work inevitably dissipates, is lost (does not disappear, but turns into an unusable form).
Thermodynamic processes
Transitions between the energy states of an ideal gas can have a different nature of changes in one or another of its parameters. Internal energy in transition processes of different types will also behave differently. Let us briefly consider several types of such processes.
- The isochoric process proceeds without changing the volume, therefore, the gas does not perform any work. The internal energy of the gas changes as a function of the difference between the final and initial temperatures.
- The isobaric process occurs at a constant pressure. The gas does the work, and its thermal energy is calculated in the same way as in the previous case.
- The isothermal process is characterized by a constant temperature, and, therefore, the thermal energy does not change. The amount of heat received by the gas goes entirely to the completion of the work.
- An adiabatic or adiabatic process takes place in a gas without heat transfer, in a thermally insulated tank. Work is done only at the expense of thermal energy: dA = - dU. With adiabatic compression, the thermal energy increases, with expansion, it decreases accordingly.
Various isoprocesses underlie the operation of heat engines. So, an isochoric process takes place in a gasoline engine at extreme positions of the piston in the cylinder, and the second and third strokes of the engine are examples of an adiabatic process. In the production of liquefied gases, adiabatic expansion plays an important role - due to it, gas condensation becomes possible. Isoprocesses in gases, the study of which can not do without the concept of the internal energy of an ideal gas, are characteristic of many natural phenomena and find application in various branches of technology.