Biological membranes are characterized by their semi-permeable nature, meaning they selectively allow certain molecules to pass while restricting others. The ability of a molecule to cross a membrane freely depends on its specific properties. Molecules that can diffuse across a membrane without the need for protein facilitation typically share three key characteristics: they are small in size, uncharged (with a net charge of 0), and nonpolar or hydrophobic. Examples of such molecules include oxygen gas (O₂), carbon dioxide gas (CO₂), and nitrogen gas (N₂), which can easily traverse the membrane due to their favorable properties.

In contrast, molecules that cannot freely diffuse across membranes exhibit opposite characteristics. These molecules tend to be large, charged (with a positive or negative net charge), and polar or hydrophilic. For instance, sugars, ions, and amino acids fall into this category, as they require protein facilitation to cross the membrane. The presence of a red arrow in diagrams often indicates that these molecules are unable to penetrate the membrane without assistance.

Additionally, there are molecules that possess mixed features; they may be small and uncharged but also polar, which allows for some diffusion, albeit at a slower rate. Water, steroids, and glycerol are examples of such molecules, which can cross the membrane but not as efficiently as the small, nonpolar molecules.

Lastly, large macromolecules, such as polypeptides, polysaccharides, and nucleic acids (DNA and RNA), are also unable to cross the membrane freely due to their size. These molecules are represented by red arrows bouncing off the membrane, indicating their inability to diffuse without protein assistance.

In summary, the ability of molecules to cross biological membranes hinges on their size, charge, and polarity. Small, uncharged, and nonpolar molecules can diffuse freely, while large, charged, and polar molecules require facilitation from proteins to cross the membrane effectively.