I want to burn firewood to save on heating costs. Burning wood releases CO2 but so does a power plant burning coal to make electricity to heat my house. Can you quantify the tradeoff? Which is better?

Coal and wood are energy sources that essentially have the same origin – Energy from the sun, that has been converted into chemical potential energy, stored in the bonds of carbohydrates through the process of photosynthesis.  In both cases, this potential energy is released through combustion. Combustion either directly generates heat, as with wood in a fireplace in a home, or indirectly generates heat, as with coal in a power plant that provides electricity to fuel an electric furnace in a home.  Both wood and coal release carbon dioxide as a by-product of the combustion process.  The primary difference is that firewood is a renewable resource whereas coal is non-renewable.  There is a finite quantity of coal in the world and the current rate of coal depletion far outpaces the rate of coal creation, which takes millions of years. 

The rate of carbon dioxide production from our current levels of coal consumption exceeds natural sinks for carbon dioxide (plants, soil, oceans), leading to its accumulation in the atmosphere, There is scientific evidence that this accumulation of carbon dioxide is altering global climate in ways that are harmful to people and the environment.  In contrast, trees used for firewood can be renewably planted, grown, and harvested.  In this way, a substantial portion of the carbon dioxide released to the atmosphere during firewood combustion is removed from the atmosphere by trees growing (photosynthesis) in the next generation.

The upside is that burning firewood is a good way to utilize dead and storm-damaged trees that might otherwise be sent to the landfill or left to rot.  Because fossil fuels are typically consumed in the planting, cultivation, harvest, and transport of firewood, it is not a carbon neutral energy source. If proper practices are not followed, there are also potential negative by-products from firewood production. These include soil erosion, sedimentation of streams, and production of particulate air pollution (soot). So, while the carbon and ecological footprint of firewood may be smaller than coal, there are still environmental impacts from firewood use. 

In terms of heat output – or energy potential, the calorific value that is the normal heat output measurement – the firewood and coal compared

Coal. Good quality anthracite offers about 25 – 26 KJ/kg (Kilojoules per Kilogram)
Firewood offers about 9 – 13 KJ/kg depending on the type of wood and how dry it is (less than half the energy of coal).

In terms of carbon emissions, you can find information about clearer burning and use of fuelwood from burn-wise online and also explore Energy Units and Calculators for additional energy comparison information.