Thermodynamics - Introduction (Under Construction)
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IntroductionThermodynamics is usually introduced by formulating equations applicable under ideal conditions - conditions which are often far removed from the conditions that are likely to be met in real-life situations, e.g. pistons moving so slowly that the fluid temperature can be maintained via conduction.
Like all ideal situations, it is hoped that real-life situations can be described approximately by the equations of the ideal situation.
In this context, the concept of a reversible process is introduced, i.e. a process in which there is no loss to friction, or viscous dissipation, etc..
Ultimately, formulae are derived for two separate scenarios
- Isothermal conditions where processes take place at constant temperature, i.e. in a perfectly conducting environment where heat can be added to/subtracted from the system in order to maintain the temperature at a constant value.
- Adiabatic conditions where processes take place in a perfectly insulated system where no heat enters or leaves the system.
The isothermal and adiabatic conditions form the 'limits' of the real situation, as illustrated in the pressure/volume diagram - note the adiabatic line is always steeper than the isothermal. We would expect a real-life situation to be somewhere between these two limits.
One other important concept is that of a perfect gas, i.e. one in which the constituent molecules are assumed to have negligible volume, and in whch there are no intermolecular forces.
Isothermal ProcessesA process in which the temperature is kept constant.
In the case of a perfect gas, the gas will satisfy Boyle's Law
p α ρ
p = kρ
Adiabatic ProcessesProcesses in which there is no heat exchange between the material and its surroundings.
First Law of ThermodynamicsThe First Law of Thermodynamics is, effectively, another name for the Law of Conservation of Energy but formulated only in terms of the quantities we are likely to meet in thermodynamics - namely heat (Q), work (W) and the internal energy of the fluid (U).
It usually stated mathematically, in differential form, as
dQ = dU + dW
where dQ has been considered positive if heat is added to the system, and dW has been considered positive if work has been done by the system, i.e. the fluid expands (if work has been done on the system, i.e. 'negative' work, it will be compressed).
So for example, heat applied to a system will
- increase the internal energy of the system
- cause the system to do work
Consider a piston - the work done by a system is given by (using work equals force times distance)
dW = F dx
where F is the external force against which the system does work, and dx is the distance moved by the point of application of this external force.
If we introduce the condition that the process is such that the pressure exerted remains the same, then since Pressure times Area equals Force
dW = F dx = pA dx = p dV
where V is the volume expansion of the fluid
So, under this assumption, the law can be expressed as
The specific heat is defined as
By substitution of an expaansion of the differential for Q, the specific heat at constant volume is given by
For a perfect gas, the specific heat at constant volume is given by
Using the same substitution for the differential, but with a bit more complicated mathematics, the specific heat at constant pressure is given by
For a perfect gas, using the equation of state for a perfect gas, the specific heat at constant pressure is given by
where R is the gas constant.
From the point of view of mathematics, thuis situation can be described by a potential whose value is a function of position, and whose differential would need to be an exact differential. Here the potential is called a function of state.
dQ is not an exact differential, but if we define
we can show that this is an exact differential - s is called entropy and is a function of state for a perfect gas.
The differential can be shown to be
which can be immediately integrated to give
Flows for which s is constant are called isentropic. From above