There is no such equation for liquid carbon dioxide extinguisher. There are three things required for a fire known as 'the fire triangle', that are
-fuel
-air (oxygen)
-heat
If we are able to cut off even one element from these three then the fire can be extinguished. Liquid carbon dioxide does exactly the same- it replaces oxygen from the burning site and since molecular oxygen is no longer available, the fire gets extinguished. Liquid carbon dioxide is kept as highly pressurised inside the extinguisher. As the gas comes out, the pressure decreases rapidly and so does the temperature of the gas and it cuts off the heat of the fire thereby extinguishing it.