  In this paper, we solve constructively the piston problem for one-dimensional
isentropic Euler equations of modified Chaplygin gas. In solutions, we prove
rigorously the global existence and uniqueness of a shock wave separating
constant states ahead of the piston when the piston pushed forward into the
gas. It is quite different from the results of Chaplygin gas or generalized
Chaplygin gas in which a Radon measure solution is constructed to deal with
concentration of mass on the piston. When the piston pulled back from the gas,
we strictly confirm only the first family rarefaction wave exists in front of
the piston and the concentration will never occur. In addition, by studying the
limiting behavior, we show that the piston solutions of modified Chaplygin gas
equations tend to the piston solutions of generalized or pure Chaplygin gas
equations as a single parameter of pressure state function vanishes.
