What is the effect of an indirect tax on the cigarette market?

Cigarettes are considered a demerit good since its consumption has a negative impact on the consumer. The product then results in an external cost as it becomes a cost for a third party as a result of an economic transaction. Thus, the MPB of cigarette consumption is greater than the MSB which proves the cigarette market to be a negative externality of consumption. 
In an attempt of ‘internalising the externality’, a government can decide to tax cigarettes by a certain amount, let's say 2.5 US$ (P/unit + 2.5$). It is therefore an indirect specific tax as it is implemented by the government on a good and is a tax per unit. The tax increases the MPC and shifts private supply to the left. This has the beneficial effect of reducing consumption of cigarettes to the optimum level of Q*. The supply curve for cigarettes shifting vertically upwards by the amount of the per-unit tax from S ( to Stax. By doing so, the cigarette consumers and producers are involved in paying the costs to society instead of the third party. MPC shifts back to intersect with Q* (S - Stax), decreasing the amount sold and raising the market price to Ptax. This brings the market from Ptax and Qtax in line with the optimal amount of output at Q*.