No, electrons do not orbit around the nucleus of an atom in the same way that planets orbit around the sun. The concept of electrons orbiting the nucleus in well-defined paths, similar to planets in the solar system, is an outdated model known as the Bohr model. According to modern quantum mechanics, electrons in atoms are described by wave functions and exist in regions of space called atomic orbitals.
In the quantum mechanical description, electrons are not point-like particles with defined positions and trajectories. Instead, they exhibit wave-particle duality and possess both particle-like and wave-like properties. The behavior and location of electrons are described by wave functions, which give the probability distribution of finding an electron in a particular region around the nucleus. The electron cloud represents the probability of finding an electron in a specific orbital.
Regarding gravity, the gravitational force between an electron and the nucleus is indeed present. However, the gravitational force is extremely weak compared to the electromagnetic forces between electrons and the positively charged nucleus. The electromagnetic forces are much stronger and dominant at atomic scales, governing the behavior and interactions of electrons within atoms.
In summary, electrons in atoms do not follow classical orbital paths and are described by quantum mechanics. While the gravitational force exists, it is negligible in comparison to the electromagnetic forces that dominate the behavior of electrons in atoms.