  We model the intracluster medium as a weakly collisional plasma that is a
binary mixture of the hydrogen and the helium ions, along with free electrons.
When, owing to the helium sedimentation, the gradient of the mean molecular
weight (or equivalently, composition or helium ions' concentration) of the
plasma is not negligible, it can have appreciable influence on the stability
criteria of the thermal convective instabilities, e.g., the heat-flux-buoyancy
instability and the magnetothermal instability (MTI). These instabilities are
consequences of the anisotropic heat conduction occurring preferentially along
the magnetic field lines. In this paper, without ignoring the magnetic tension,
we first present the mathematical criterion for the onset of composition
gradient modified MTI. Subsequently, we relax the commonly adopted equilibrium
state in which the plasma is at rest, and assume that the plasma is in a
sheared state which may be due to differential rotation. We discuss how the
concentration gradient affects the coupling between the Kelvin--Helmholtz
instability and the MTI in rendering the plasma unstable or stable. We derive
exact stability criterion by working with the sharp boundary case in which the
physical variables---temperature, mean molecular weight, density, and magnetic
field---change discontinuously from one constant value to another on crossing
the boundary. Finally, we perform the linear stability analysis for the case of
the differentially rotating plasma that is thermally and compositionally
stratified as well. By assuming axisymmetric perturbations, we find the
corresponding dispersion relation and the explicit mathematical expression
determining the onset of the modified MTI.
