Integration
Definition
... the area under a curve or line, bounded by limits.
Where it occurs.
In closed systems the reversible, moving boundary work follows from the integral,
$$ W = -\int_1^2 p dV $$
( This applies to the IUPAC 1992 convention. )
Special features
There are analytical solutions for isobaric, isothermal, isentropic and polytropic processes.
How to draw it
- Click the radio button by "vertical construction lines"
- Use the sliders to move the two construction lines to the start volume and end volume. (Note - the position of the construction line can be finely adjusted with the keyboard curser keys. )
- Identify a a point of interest > . Plot a curve .
- Press "integrate curve". Values of work, change in internal energy and heat transfer are reported immediately above the plot area.
A special case is when the curve is very steep and the volume change is barely visible. Here one can use horizonal construction lines to describe limits in
terms of pressure.
The theory
The integral is found from the trapezium rule .
If one assumes an ideal gas then the change in internal energy follows from
$$ \Delta U = U_2 - U_1 = m c_v (T_2 - T_1) $$
In a closed system, assuming no changes in kinetic energy and potential energy, the Non-Flow Energy Equation is
manipulated to yield heat transfer,
$$ Q = \Delta U - W $$
Exercises
... to follow.
Links
... to follow.