Negative Externality with Tax and Free TRade I am unsure of how to think about this negative externality question when there is both tax and free trade. 

A country is producing plastic, but it has a negative externality cost of 4 dollars/bottle. The demand is $Q_D=12-P$ and the supply is $Q_S=P$. If a 4 dollars/bottle tax is enacted and the country is opened up to free trade with a world price of 4 dollars, what would the total social surplus be? Based on my calculations, it would be 24 dollars, just the same as when the country has no tax and is open to free trade. Is this correct thinking? Or am I not thinking about the tax and free trade correctly? Thanks.
Rearranging your demand and supply equations gives us:
$$P=Q_s\quad and\quad P=12-Q_D$$

Adding the tax on the supply side gives us:
$$P=Q_s+4$$

Solving for equilibrium, we get $P=8$ and $Q=4$

However, we did not necessarily need to solve for this equilibrium because the Supply curve shifted left, and the new intercept is at $P=4$. This implies that no plastic will be produced domestically because the world price will be lower at any positive level of production. 

To solve for total surplus, we need only solve for consumer surplus because no production will occur domestically. We need to solve for $Q_D$ when $P=4$.
$$4=12-Q_D\implies Q_D=8$$
$$\implies TS=CS=\bigg(\frac{1}{2}\bigg)*8*8=32$$

Is this the same as when there is no trade and no tax?

$$P=Q_s\qquad P=12-Q_D$$

Equilibrium yields $P=Q=6$
$$TS=CS+PS-DWL$$
$$TS=CS=\bigg(\frac{1}{2}\bigg)*6*6$$

To calculate $DWL$, we split the cost into private cost and social cost. When we do this
$$PC=Q\qquad SC=Q+4$$

The values we need to calculate the $DWL$ triangle end up being $Q=4$, $P_{PC}=4$ and $P_{SC}=8$

$$DWL=\bigg(\frac{1}{2}\bigg)*4*4=8$$

$$\implies TS=18+18-8=32$$

Your intuition was correct that the total surplus is the same in both cases.