Adiabatic process, its essence and formulas
An adiabatic process (referred to as adiabatic in some sources) is a thermodynamic process that occurs in the absence of heat exchange with the environment. There are several factors that characterize this class. For example, the adiabatic process occurs dynamically and is laid down in a short period of time. There are processes of this class, as a rule, instantly.
Connection with the first law of thermodynamics
The adiabatic process (adiabatic) can be directly related to the first law of thermodynamics. The wording “by default” is as follows: a change in the amount of heat in the system during a thermodynamic process in it will be numerically equal to the sum of the change in the internal energy of an ideal gas and the work done by this gas.
If we try to write the first law of thermodynamics in its standard form, we get the following expression: dQ = dU + dA. And now we will try to modify this formula in relation to the adiabatic process.As mentioned earlier, such processes occur under the condition that there is no heat exchange with the surrounding (external, as some literary sources call it) medium.
In this case, the formula describing the first law of thermodynamics will take the following form: dA = -dU. Now a little more about modification. If we say that heat transfer in the system does not occur, the change in the amount of heat (indicated in the formula of the first law of thermodynamics by dQ) will be zero. Consequently, we can transfer one of the addends from the right to the left, after which we obtain the formula reduced to the form described earlier.
Corollary of the first law of thermodynamics for the adiabatic process
Suppose that an adiabatic process has occurred in the system. In this case, it is possible, without going into the smallest details, to say that the gas does the work during expansion, but at the same time it loses its internal energy. In other words, the work performed during adiabatic expansion of gas will be carried out due to the loss of internal energy. Therefore, as an outcome of this process, we will consider a decrease in the temperature of the substance itself.
It is absolutely logical to assume that if the gas is adiabatically compressed, its temperature will rise. It is easy to see that during the process all the main characteristics of an ideal gas will change. It is about its pressure, volume and temperature. Therefore, the name of the adiabatic process by the isoprocess became a gross mistake.
Adiabatic process. Formulas
Earlier, a formula derived from the first law of thermodynamics was written down. Using it, we can easily calculate the work in general, which the gas will perform during the adiabatic process. As you might have guessed, we will do this with the help of integration.
So, in order to obtain the general formula for the work for x moles of gas, we integrate the expression of the first law of thermodynamics for an adiabatic process. All this will look like this: A = - (integral) of dU. Expand this expression, we get: A = - xCv (integral from T1 to T2) dT.
Now that we have brought the integral to a finite form, we can simplify it. At the output, we obtain the following formula: A = - xCv (T2 - T1). Well, the last step will be a slight simplification. We will get rid of the minus before the formula.To do this, we make a small permutation in brackets, changing the final temperature with the initial places. As a result, we obtain: A = xCv (T1 - T2).
Using the first law of thermodynamics for an adiabatic process, we can find the adiabatic equation. In this case, it will be recorded for an arbitrary number of moles of an ideal gas. So, we write down the original formula. It looks like this: dA + dU = 0. But we know perfectly well that the work of an ideal gas is numerically nothing more than the product of pressure and volume change.
At the same time, the change in internal energy will be equal to the work taken with the opposite sign. And we have already found it with the help of integration. Therefore, the first law of thermodynamics for the adiabatic process can take the following form: pdV + xCvdT = 0. We need to exclude one indicator from this equation, namely, temperature. Rather, its changes. To do this, we turn to the equation that is quite often used in molecular physics. Namely, the Mendeleev-Clapeyron equation.
We need to differentiate it, which we will do. So, in general, the equation is as follows: PV = XRT.Due to the differentiation, it will be reduced to the following form: pdV + Vdp = xRdT. From here we can express the change of energy. It will be equal to the left part, divided by the product of the amount of a substance and the universal gas constant. In other words, the formula would be: (pdV + Vdp) / xR. It remains only to simplify it. As a result, we obtain the following expression: dT = (pdV + Vdp) / x (Cp - Cv)
In fact, the first part of the task is completed. It remains only to bring everything to mind.
Secondary expression Value substitution
Let us take the Mendeleev-Clapeyron formula obtained as a result of differentiation and substitute it into the expression we previously derived for the first law of thermodynamics with respect to the adiabatic process. So what do we get? All this cumbersome expression takes the following form: pdV + xCv ((pdV + Vdp) / x (Cp-Cv)) = 0.
To simplify all this, we must take into account a couple of facts. First, the expression can be simplified by reducing it to a common denominator. When we get one fraction, we can use the good old rule that says that a fraction is zero when its numerator is zero, and the denominator is non-zero. As a result of the combination of all these actions, we get the following expression: pCpdV - pCvdV + pCvdV + VCvdp = 0.
Now the next step is to divide this expression into pVCv.We obtain the sum of two parts, giving as a result zero. This will be Cp / Cv * dV / V + dp / p = 0. This formula must be integrated. Then we get the following expression: y (integral) dV / V + (integral) dp / p = (integral) 0.
Well, then everything is quite simple. Using the integration formulas (table integrals can be used to make everything simpler), we end up with the following entry: y ln V + ln p = ln (const). It turns out that p (V) y = const. In molecular physics, this expression is called the Poisson equation. Many scientific literature sources also call this formula the adiabatic equation. At the same time, the value of y, which occurs in this record, is called the adiabatic index. It is equal to (i + 2) / i. It should be noted that the adiabatic index is always greater than one, which, in principle, is logical.
Examples of adiabatic processes
Soon after the adiabatic process was discovered, a huge number of different studies started. Thus, the first theoretical model was created that is related to the Carnot cycle. It was she who allowed the establishment of conditional limits that restricted the development of heat engines. But in the case of some real processes, it is rather difficult to carry out the Carnot cycle. The thing is that it consists of isotherms.And they, in turn, require the specification of a certain rate of thermodynamic processes.
In order to circumvent such problems, the Otto cycle was invented, as well as the gas liquefaction cycle. They have become widely used in solving specific problems in practice. Starting studies have shown the possibility of describing some natural processes in the adiabatic plane, which made it possible to identify the general patterns of the corresponding processes. An example of an adiabatic process can be called a chemical reaction that takes place inside a certain volume of gas, if the system is closed and there is no heat exchange with the external environment.