Work done under constant pressure Work done under a constant pressure.




I know $$\Delta U = \Delta V \times \pi_T + C_V\Delta T $$

I thought I can give a try to find work done from above equation under constant pressure for an ideal gas. 

$$q_p + w_p = C_V\Delta T$$
$$\implies C_p\Delta T + w_p = C_V \Delta T$$

$$\implies w_p = -\Delta T (C_p - C_V)$$
$$\implies w_p = - R\Delta T $$

Adiabatic work is $w_\mathrm{ad} = C_V \Delta T$

For a adiabatic work under constant pressure $w_\mathrm{ad} = w_{q} \implies C_V = -R \implies C_p = 2R$

Constant $C_p, C_V$ and negative $\gamma$ does not look correct to me.




Which of my steps are incorrect ?
You can't do adiabatic reversible work at constant pressure.  The pressure has to change.  If you want to do adiabatic irreversible work at constant pressure, say by suddenly increasing or decreasing the pressure load to begin with, and then holding it constant during the process, this is sometimes referred to as adiabatic work at constant pressure, but it really isn't.  The final pressure will not be equal to the initial pressure that was present before you changed the applied load.