Work done in an isothermal irreversible process In calculating the work done in an isothermal irreversible process, why do we consider the external pressure only (using formula: $W_{\text{ext}} = P_{\text{ext}} \mathrm{d}V)$ but not the internal pressure? When the system expands, the internal pressure does the positive work, but even in this scenario, we use the equation as mentioned above. Why is this happening?
The key word here is irreversible. Irreversible processes are more like sudden jerks or a quick change in the system and therefore the ${P_{\text{ext}}}$ is not same as the ${P_{\text{int}}}$ throughout the process.


  Why do we NOT use the internal pressure (${P_{\text{int}}}$) to find the work done?
  It is very difficult to find the ${P_{\text{int}}}$ exerted by the gas in the container since ${P_{\text{int}}}$ is varying with time. For this, we would have to use integration. Something like this:$${W_{\text{int}} = P(V)_{\text{int}}\mathrm{d}V}$$
  (where $P(V)_{\text{int}}$ is a function of $V$) 
  But we know the value of ${P_{\text{ext}}}$ and this is constant throughout the process. This is why we use ${P_{\text{ext}}}$ instead of troubling ourselves with integration. Something like this: $$W_{ext}= P_{\text{ext}} \mathrm{d}V$$


So if a system undergoes a decrement in volume suddenly, it is the external pressure that does the positive work throughout this decrement. The equation would be: $$W_{\text{ext}}= P_{\text{ext}}\mathrm{d}V$$ (However we can use ${P_{int}}$ as well, please read the NOTE at the bottom)

And if a system undergoes an increment in volume suddenly, ${P_{\text{int}}}$ is doing the positive work. However, to find the $W_{int}$ we would have to use integration since ${P_{\text{int}}}$ is varying with time. It is much easier if we just find the negative work done by ${P_{\text{ext}}}$ and since we know that $${W_{\text{int}} = - W_{\text{ext}}}$$ we can find the work done. This is why we consider the ${P_{\text{ext}}}$ and according to the situation in the question, we apply the signs. 

NOTE: Magnitude of work done by the internal pressure and external pressure is always same. We only use ${P_{\text{ext}}}$ because it is constant whereas ${P_{\text{int}}}$ varies with time. If we use integration to find ${W_{\text{int}} = P(V)_{\text{int}}\mathrm{d}V}$, where $P(V)_{\text{int}}$ is a function of $V$, we will get the same magnitude as we would from $W_{\text{ext}}= P_{\text{ext}} \mathrm{d}V$. Only signs are opposite.