Thermodynamic properties of the new multiferroic material (NH$_4$)$_2$[FeCl$_5$(H$_2$O)]

(NH$_4$)$_2$[FeCl$_5$(H$_2$O)], a member of the family of antiferromagnetic $A_2$[Fe$X_5$(H$_2$O)] compounds ($X$ = halide ion, $A$ = alkali metal or ammonium ion) is classified as a new multiferroic material. We report the onset of ferroelectricity below ~6.9 K within an antiferromagnetically ordered state ($T_N \sim 7.25 K$). The corresponding electric polarization can drastically be influenced by applying magnetic fields. Based on measurements of pyroelectric currents, dielectric constants and magnetization we characterize the magnetoelectric, dielectric and magnetic properties of (NH$_4$)$_2$[FeCl$_5$(H$_2$O)]. Combining these data with measurements of thermal expansion, magnetostriction and specific heat, we derive detailed magnetic field versus temperature phase diagrams. Depending on the direction of the magnetic field up to three different multiferroic phases are identified, which are separated by a magnetically ordered, but non-ferroelectric phase from the paramagnetic phase. Besides these low-temperature transitions, we observe an additional phase transition at ~79 K, which we suspect to be of structural origin.