The first law of thermodynamics does not forbid heat engines with 100% thermodynamic efficiency; it is the second law of thermodynamics which states that heat engines cannot obtain 100% efficiency. Can the first law ever show this second law result? Yes. We show that the first law alone can show that some heat engines cannot achieve efficiencies of 100% when self-contained cycles (which we define) are considered for some temperature reservoirs. Self-contained cycles do not rely on the surroundings for anything but heat transfer to the cycle. Self-contained cycles require that any work input required by the cycle must come from previously stored energy from the cycle (such as, by potential energy in a weight previously lifted by the engine when performing work). Also, any regenerative heat output must occur before it is used as regenerative heat input. Using an example of a self-contained modified Carnot cycle that utilizes an ideal gas, the first law shows that efficiencies of 100% are not possible for T[subscript H] < 2 T[subscript C], where TH is at a hotter temperature than T[subscript C], a colder temperature. These cycles, which could operate through a finite temperature difference ??? in some regions of a PV diagram, are not possible in other regions.