The propagation of Rayleigh surface waves in an isotropic thermoelastic solid half-space is the focus of the current study, which takes into account the compact form of six distinct thermoelasticity theories. An isothermal boundary surface in the absence of tangential and normal stress is used to solve the problem. A dispersion equation with irrational terms is obtained after creating a mathematical model. This equation needs to be transformed into a rational polynomial equation in order to use the algebraic method to find exact complex roots. The roots are filters for in-homogenous wave propagation that decays with depth. Then these roots are used to compute the numerically characteristic properties of the Rayleigh wave, which include phase velocity, attenuation coefficient, and polarisation of particles. The results are presented graphically for particular cases of thermoelasticity by using the physical data of copper metal.