5 0 0.2 0.4 0.6 0.8 1 4 8 12 16 T c  (MeV) $_{v}$ = 0.00 $_{v}$ = 0.02 $_{v}$ = 0.03 FSUGold phase can prevail for larger values of T$_{c}$ . We find that for the soft symmetry energy (Λ$_{v}$ = 0 . 03) even pure neutron matter ( α = 1) can exhibit LGP transition at T ≤ T$_{c}$ = 2 MeV. While the stiffest symmetry energy (Λ$_{v}$ = 0) at α > 0 . 9 predicts that the matter can only be in the pure gas phase at all temperatures. In summary the effects of isospin symmetry interaction on the liquid-gas phase transition in hot neutron-rich nu- clear matter is investigated. For this we have used the two accurately calibrated relativistic mean field models, the NL3 [27] and the FSUGold [28] wherein the density dependence of nuclear symmetry energy at zero tempera- ture has been constrained within a limited range by neu- tron skin thickness data of several atomic nuclei. We find considerable sensitivity of the symmetry energy on the features of phase transition. Softer symmetry en- ergies give progressively larger phase-separation bound- aries with higher critical values for pressure and isospin asymmetry as well as maximal asymmetries. At a given asymmetry we find the critical temperature for the exis- tence of the mixed liquid-gas phase increases with softer symmetry energy and predicts the possible occurrence of even an unstable pure neutron matter at finite tempera- tures. couplings Λ$_{v}$. For symmetric nuclear matter ( α = 0), the critical temperature for LGP transition in this model is T$_{c}$ = 14 . 7 MeV. With increasing asymmetry α > ∼ 0 . 6, T$_{c}$ decreases rapidly. A softer density dependence in sym- metry energy (larger Λ$_{v}$) shows the coexisting liquid-gas FIG. 5: The critical temperature T c versus isospin asymmetry α for different Λ v in the FSUGold set. [1] J. Pochodzalla et al.; Phys. Rev. Lett. 75 (1995) 1040. [2] J.B. Elliot et al.; Phys. Lett. B 418 (1998) 34. [3] J.P. Bondorf, A.S. Botvina, A.S. Iljinov, I.N. Mishustin, and K. Sneppen, Phys. Rep. 257 (1995) 130. [4] D.Q. Lamb, J.M. Lattimer, C.J. Pethick, and D.G. Ravenhall, Phys. Rev. Lett. 41 (1978) 1623. [5] G.F. Bertsch and P.J. Siemens, Phys. Lett. B 126 (1983) 9. [6] H. Jaqaman, A.Z. Mekjian and L. Zamick, Phys. Rev. C 29 (1984) 2067. [7] S.J. Lee and A.Z. Mekjian, Phys. Rev. C 63 (2001)044605. [8] J.P. Blaizot, Phys. Rep. 64 (1980) 171. [9] K.A. Brueckner, S.A. Coon, J. Dabrowski, Phys. Rev. 168 (1967) 1184. [10] P.J. Siemens, Nucl. Phys. A 141 (1970) 225. [11] L.W. Chen, C.M. Ko, and B.-A. Li, Phys. Rev. Lett. 94 (2005) 032701; Phys. Rev. C 72 (2005) 064309. [12] B.-A. Li, L.-W. Chen and C.M. Ko, Phys. Rep. 464 (2008) 113. [13] V. Baran, M. Colonna, V. Greco and M. Di Toro, Phys. Rep. 410 (2005) 335. [14] P. Danielewicz, R. Lacey and W.G. Lynch, Science 298 (2002) 1592. [15] D.V. Shetty, S.J. Yennello and G.A. Souliotis, Phys. Rev. C 76 (2007) 024606. [16] M. Centelles, X. Roca-Maza, X. Vi˜ nas, and M. Warda, Phys. Rev. Lett. 102 (2009) 122502. [31] H.S. Xu et al., Phys. Rev. Lett. 85 (2000) 716. [30] H.R. Moshfegh and M. Modarres, Nucl. Phys. A 792 (2007) 201. [29] W. Zhuo, Z.H. Li, U. Lombardo, G.C. Lu and H.J. Schulze, Phys. Rev. C 73 (2006) 035208. [28] B.G. Todd and J. Piekarewicz, Phys. Rev. Lett. 95 (2005) 122501. [27] G.A. Lalazissis, J. K¨ onig, and P. Ring, Phys. Rev. C55 (1997) 540. [26] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [25] C.J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86 (2001) 5647; Phys. Rev. C 66 (2002) 055803. [24] B.A. Brown, Phys. Rev. Lett. 85 (2000) 5296. [23] P. Wang, Phys. Rev. C 61 (2000) 054904; P. Wang, D.B. Leinweber, A.W. Thomas and A.G. Williams, Nucl. Phys. A 748 (2005) 226. [22] W.L. Qian, R.K. Su and P. Wang, Phys. Lett. B 491 (2000) 90. [21] H. M¨ uller and B.D. Serot, Phys. Rev. C 52 (1995) 2072. [20] J. Xu, L.-W. Chen, B.-A. Li and H.-R Ma, Phys. Lett. B 650 (2007) 348. [19] J. Xu, L.-W. Chen, B.-A. Li and H.-R. Ma, Phys. Rev. C 75 (2007) 014607. [18] D.J. Dean, K. Langanke, and J.M. Sampaio, Phys. Rev. C 66 (2002) 045802. [17] B.K. Sharma and S. Pal, Phys. Lett. B 682 (2009) 23.