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Before diving into the effects of carbon, it is important to comprehend the concept of a metallic lattice. Metals have a structure in which positively charged ions (cations) are arranged in a regular pattern along with free electrons that can move throughout the lattice. This unique structure gives metals their characteristic properties, such as high electrical and thermal conductivity, malleability, and ductility.

When carbon atoms are introduced into a metallic lattice, they typically occupy the interstitial positions (spaces between the metal ions). Carbon atoms are relatively small compared to metal ions, and thus they can easily fit into these spaces and form a mixture of metal and carbon atoms known as an interstitial alloy.

The interstitial carbon atoms interrupt the regular arrangement of the metal ions and cause lattice distortion. This distortion increases the resistance to dislocation motion, which is responsible for plastic deformation in metals. As a result, the metal becomes harder and less ductile. In general, as the carbon concentration increases, the dislocation motion is further impeded, which leads to even higher hardness and lower ductility.

The presence of carbon atoms in the metallic lattice also affects the electrical conductivity of the metal. The free electrons are responsible for the high electrical conductivity in metals. However, the presence of interstitial carbon atoms causes lattice distortion and reduces the regularity in the arrangement of metal ions. This distortion impedes the motion of free electrons and results in a decrease in electrical conductivity.

Similarly, the introduction of carbon into the metallic lattice reduces the thermal conductivity of the metal. The thermal conductivity of a material is dependent on its lattice vibrations (phonons) as well as free electron motion. The presence of carbon atoms in the lattice causes distortion leading to an increased resistance to the flow of heat through the lattice vibrations. Additionally, the reduced electrical conductivity due to the presence of carbon atoms also contributes to the decrease in thermal conductivity, as the motion of free electrons is responsible for transferring heat within the metal. In conclusion, introducing carbon into a metallic lattice generally results in a harder, less ductile substance with lower electrical and thermal conductivities due to the interstitial carbon atoms causing lattice distortion, impeding dislocation motion, and hindering the movement of free electrons responsible for electrical and thermal conduction.