Introduction of carbon into a metallic lattice generally results in a harder, less ductile substance with lower electrical and thermal conductivities. Explain why this might be so.

A metallic lattice is a regular arrangement of metal atoms in a crystalline structure. The electrons in the outermost shell of each metal atom are shared among all the atoms in the lattice, forming a "sea" of electrons. This arrangement not only holds the atoms together but also gives metals their unique properties such as high electrical conductivity, ductility, and thermal conductivity.

When carbon is introduced into a metallic lattice, it usually exists in the form of an interstitial impurity. Carbon atoms occupy the interstitial spaces between metal atoms. As the carbon atoms are smaller than the metal atoms, they can easily fit into the interstices without causing significant distortion of the lattice structure.

The introduction of carbon atoms in the metallic lattice increases the hardness of the material. This increase in hardness is due to the change in the lattice structure caused by the presence of the carbon atoms. Carbon atoms exert a repulsive force on the surrounding metal atoms that results in the metal atoms being pushed slightly further apart. This increased distance between the metal atoms leads to a higher resistance against the movement of dislocations, which in turn makes the material harder.

Ductility is the ability of a material to deform plastically without breaking. The presence of carbon atoms in the lattice structure hinders the smooth movement of dislocations, which is necessary for the material to deform plastically. The increased resistance against dislocations makes the material less ductile, causing it to be more brittle and prone to breaking.

Electrical conductivity is the ability of a material to allow the flow of electric current. Due to the presence of carbon atoms in the interstitial spaces, the flow of electrons in the "sea" of electrons within the metallic lattice is impeded. The carbon atoms effectively act as scattering centers, causing the electrons to deviate from their paths and, as a result, reduce the electrical conductivity of the material.

Thermal conductivity is the ability of a material to transfer heat. Similar to the decrease in electrical conductivity, the presence of carbon atoms obstructs the flow of heat through the material. The lattice vibrations, known as phonons, which are the primary carriers of thermal energy, are scattered at the sites where carbon atoms are present. This scattering of phonons reduces the material's thermal conductivity. In summary, the introduction of carbon atoms into a metallic lattice increases the hardness and decreases ductility, electrical conductivity, and thermal conductivity. This phenomenon can be attributed to the presence of carbon in the interstitial spaces and the effect it has on the lattice structure, dislocations, electron flow, and phonon scattering.