A simple way of describing this would be using the following equation: $$\frac{\Delta C_{N - 1}}{\Delta T} = k_{\frac{1}{T}}$$ This can be rearranged to yield $$\frac{\Delta C_{N}}{\Delta T} = \frac{k_{\frac{1}{T}}}{\frac{1}{T}}$$ $$\frac{\Delta C_{N}}{\Delta C_{N - 1}} = \frac{k_{1}}{\frac{1}{T}}$$ $$\frac{k_{2}}{k_{1}} = \frac{T_{2}}{T_{1}}$$ And hence the definition of 'isothermal compression' or 'isothermal expansion' of a substance. The idea of an adiabatic process was simply to say that when an object is warmed in a gas, the heat must flow from the body to the surrounding gas in order for the gas to remain at a constant temperature. A 'heat sink' in this sense would have to be created which exactly balances the rate of heat flow into the body. And is this really the same thing as doing work on an object? Well, that would come from the fact that a force acts over some distance, which means that the work $W$ done by a force would be defined as $$W = \int F.d\ell$$ And this means that a net force on an object, to do work on an object is defined as $$\int \vec{F} \bullet d\vec{S}$$ So how would this all translate to the notion of enthalpy? Well, again this would be used in a similar way to 'heat', however the term would be synonymous with the work done by the system. Therefore, a constant enthalpy would define an adiabatic process, however, in the case of an adiabat the change in enthalpy would still occur at constant temperature, but the internal pressure of the substance remains constant too! This means that any adiabatic change would have to occur at a constant pressure! The isentropic process was also used to solve this by stating that any process where $C_{p} \bullet dT$ is constant would also be adiabatic (again using an isothermal compression/expansion). That being said, this process is only adiabatic if the system pressure is conserved as well. This would require that $P \bullet dV$ was also a constant, and therefore $\frac{dP}{dV}$ had to remain constant. It can be shown that this would always be the case if $C_{p} \bullet dT$ was constant (because of the ideal gas law). This is clearly not a problem in an isentropic process (however one cannot do this in a finite region). Therefore, a change in temperature in an isentropic process must be such that the entropy $\frac{dS}{dT}$ also remains constant. And this of course would be the same as $\frac{C_{p}}{T}$ also remaining constant. And again, this constant would be the one defined in an adiabatic process. Therefore, in a simple sense, the isentropic processes could be seen as adiabatic processes with a constant entropy, rather than just with a constant temperature, where temperature was only defined as a function of pressure. In a similar manner to the isentropic process, the process would not be adiabatic if the heat flow $dQ$ was not constant. This would be in essence a heat engine, such that the heat engine $dQ$ was defined by a heat capacity $C_{q}$ of the machine. By knowing that heat flow is defined as $$dQ = \frac{C_{p} - C_{q}}{T}\cdot dT$$ We can show that this heat flow $dQ$ would be equivalent to the work done by the system $dW$. Therefore, any form of heat flow would have to be $\frac{C_{p} - C_{q}}{T}$ in order for this process to be adiabatic. In a similar way, $P \bullet dV$ would have to be constant if this process was to be adiabatic. Again, this would require the system $C_{p} \bullet dT$ to remain constant in time. And this would be true if and only if $C_{q} \bullet dT$ also remained constant. And since heat flow is equivalent to work done, $\frac{C_{p} - C_{q}}{T}$ will be the heat flow in an isothermal process. Hence, an isothermal process can be seen as a special case of an adiabatic process with a constant heat flow, $dQ$. And the same can be said of a constant heat flow, $dQ$ with a constant entropy, $\frac{C_{p} - C_{q}}{T}$, in an isentropic process. Therefore, by looking at the problem from this view, we can indeed say that a constant pressure is what allows for an adiabatic process. And by doing this, we see that the entropy of the system must remain constant in order for the process to be adiabatic. Otherwise, we would have a paradox where the system could not be defined as adiabatic. However, if the entropy did remain constant, then the heat flow $dQ$ would have to be equal to the enthalpy change $C_{p} \bullet dT$. The entropy change would have to be equal to the heat transfer, so that a heat engine in equilibrium would be defined by $\frac{dQ}{T}$. Therefore, the processes that could be defined as adiabatic processes are those processes that maintain $dQ = C_{p} \bullet dT$ for the system. Or in other words, the isentropic processes! An ideal gas is just a system where $C_{q} \bullet dT$ is constant. Or in other words, any process where $C_{q} \bullet dT$ is constant could be regarded as an isentropic process. This would mean that any adiabatic process where $C_{q} \bullet dT$ was not constant would not be an isentropic process! This is a paradox, because it is defined that an adiabatic process is an isentropic process. Therefore, we must have that an adiabatic process in which $C_{q} \bullet dT$ was not constant would be another word for something else, rather than an adiabatic process. And this would mean that there is really no need to distinguish an isentropic process from an adiabatic process! Or in other words, we could say that for a system with a constant entropy, an adiabatic process is an isentropic process. To close: Can we say that temperature is synonymous with the coefficient of convection? It may seem like we are doing work on the object by heating it. This is due to the fact that we are changing the energy of the atoms within the system as we heat it up! This implies that temperature is synonymous with the coeffcient of converence. The reason this may seem like a paradox is that $C_{p}$ is constant in an adiabatic process. Or in other words, the system's enthalpy $H$ does not change with a change in temperature! Therefore, this makes us think that we are doing work on the object (which would cause a change in $C_{p}$). However, we can still do adiabatic work if $\frac{dH}{dT}$ is constant. This means that $\frac{dT}{dT}$ must be constant. This would be the case if $\frac{dH}{dT} \propto C_{p}$. Now if we say that temperature is synonymous with $C_{p}$, then we would have that $\frac{dH}{dT} \propto dT$. Therefore, as the temperature of the system changes, the enthalpy must change. This may seem like a paradox, however we can use this fact to create an argument that energy flows from hotter to colder systems. That being said, we can look at the following diagram: \begin{array} \text{ \large \textbf{Heat flow diagram for a system}} \end{array} In this diagram, we can see that a constant $C_{p}$ means that the temperature gradient is constant for the object. So by this we can see that in an adiabatic process, energy flows from a hotter to colder object (by heating it up). This is due to the fact that energy always flows from hotter to colder systems. If the temperature difference between objects is large enough, this would imply that the heat flow would be large enough for a significant amount of energy to be transferred. So by this we can say that $\frac{dH}{dT}$ remains constant. Hence, the adiabatic process is adiabatic and $\frac{dH}{dT}$ remains constant, if the process maintains $C_{p}$. And because $\frac{dH}{dT}$ is constant, the entropy of the system will be constant (as $\frac{dS}{dT}$ is equal to $C_{p}/T$) and hence $C_{p}$ is constant. A: Here is a simple proof for an adiabatic process being always isothermal. Assume an adiabatic process occurs at temperature $T$, and an amount of heat is taken from the body. To maintain the temperature of the object constant during the process, a certain amount of heat $\delta Q$ has to be added to the object. At the end of the process, we must still have the exact same amount of heat $\