Heat engine efficiency for four-dimensional charged AdS black holes in Rastall gravity

In this paper, we analyze the P-v criticality and the heat engine efficiency of two black holes, one is the four-dimensional charged AdS black hole in Rastall gravity and the other is the four-dimensional charged AdS black hole surrounded by quintessence in Rastall gravity. For the four-dimensional charged AdS black hole in Rastall gravity, the ratio Pc upsilon cTc is related to the Rastall parameter lambda. When lambda=0 the ratio becomes PcvcTc=38, which is same as the van der Waals liquid-gas system. Additionally, as the Rastall parameter lambda increases, the heat engine efficiency eta of the black hole will increase. Moreover, the efficiency eta decreases rapidly at first and then increases with the entropy S2. But the efficiency ratio eta eta c decreases with the Rastall parameter lambda. For the four-dimensional charged AdS black hole surrounded by quintessence in Rastall gravity, as the charge q increases, the critical specific volume upsilon c and the efficiency ratio Pc upsilon cTc also increase, however, the critical temperature Tc and the critical pressure Pc gradually decrease. Moreover, the heat engine efficiency eta and the efficiency ratio eta eta c of the black hole are similar to the four-dimensional charged AdS black hole in Rastall gravity.