We study the time-asymptotic behaviour of solutions to the Navier-Stokes equations for a one-dimensional viscous polytropic ideal gas in the half-line. Using a local representation for the specific volume, which is obtained by using a special cut-off function to localize the problem, and the weighted energy estimates, we prove that the specific volume is pointwise bounded from below and above for all x, t and that for all t the temperature is bounded from below and above locally in x. Moreover, global solutions are convergent as time goes to infinity. The large-time behaviour of solutions to the Cauchy problem is also examined.