Moving Boundary Work and Polytropics

Specific Heat

There is no latent energy stored in intermolecular bonds so internal energy is due entirely to molecular motion

$c_v=\frac{du}{dt}$ Energy required to raise the temperature of a unit mass of a substance by one degree $u_2-u_1=\overline{c_v}\left(T_2-T_1\right)$

Moving Boundary Work

$W_{1-2}={\int }_1^{2}P\left(V\right)dV$

Polytropic Process

More than one thing changing at a time (heat and work interactions change simultaneously)

These are found in pumps, turbines, compressors, etc. in many thermodynamic cycles.

$P{V}^n=C$ For the work output of a process If n != 1:
$w_{1-2}=\frac{P_2{V}_2-P_1{V}_1}{1-n}$ Or if we can assume ideal gas:
$W_b=\frac{mR\left(T_2-T_1\right)}{1-n}$
For the work output of a process if If n = 1:
$w_{1-2}=P_1{V}_1\ln \left(\frac{V_2}{V_1}\right)=P_2{V}_2\ln \left(\frac{V_2}{V_1}\right)$
Or if we can assume ideal gas:
$PV=mRT$
Which is the ideal gas law for an isothermal process.

For the work output of a process If n = 0
$w_{1-2}=P\left(V_2-V_1\right)$
This is an isobaric process.

For the work output of a process If n = infinity. $w_{1-2}=0$
which is an isochoric process

For the work output of a process If n = γ: Where
$\gamma =\frac{C_p}{C_n}$
n = 1.4 for main and this is an adiabatic process.

n

n is found by curve-fitting to experimental data where n deviates from 1.4 or 1.0 for compression amd power strokes.