Carbon emissions from burning of fossil fuels and deforestation have led to increasing concentrations of carbon dioxide in the atmosphere, which in turn lead to an increase in radiative heating. When the carbon dioxide is emitted into the atmosphere, a significant fraction of the carbon is initially taken up by the oceans, which reduces the immediate radiative heating from the emissions. Eventually, the atmosphere and ocean approach an equilibrium state after several hundred to a thousand years. The carbon dioxide remaining within the atmosphere leads to a long-term radiative heating. The problem of understanding how the carbon cycle varies is usually addressed by integrating large, complicated climate models. While these climate models are useful, we wish to adopt a different approach focussing on a long-term equilibrium state for the atmosphere and ocean. At this equilibrium state, we predict that carbon emissions lead to an exponential increase in the atmospheric concentration of carbon dioxide and a linear increase in the long-term radiative heating. If all our conventional carbon reserves are utilised, then the longterm radiative heating is 4 times larger than the present-day increase in radiative heating from carbon emissions. We aim to test this prediction in a climate model including carbon and temperature feedbacks. In a similar manner, we aim to compare our predictions of how increasing ocean stratification and acidity will affect the carbon cycle. Our longterm equilibrium solutions will be compared with climate models representing the cycling of carbon in the ocean and the whole Earth System. This comparison will provide insight into how the carbon system operates and a longterm context for climate change which policy makers need to consider when comparing different carbon emission scenarios.