An ideal gas is undergoing an isothermal expansion against a constant external pressure. Which of the following statements is true?
If the gas is ideal, for an isothermal expansion 

$$p=\frac{nRT}{V}=\frac{\textrm{constant}}{V}$$

As the volume of the system changes, so must the pressure.

Now a condition of reversibility is mechanical balance between the system and surroundings, that is, $p=p_{\textrm{ext}}$. However, if a process is performed at constant external pressure and temperature, then as shown above the condition of mechanical balance cannot be satisfied, since $p$ will change during the expansion whereas $p_{\textrm{ext}}$ is constant. Therefore the process must be irreversible.