Tank system of a motor vehicle with a volume modifying element provided in the inner chamber of the fuel tank, the so-called compensation volume of said volume modifying element being connected to the surroundings depending on boundary conditions, comprising the following features: • the inner chamber of the fuel tank is connected to the surroundings via a valve unit, which is open when the fuel tank is being filled as well as when an overpressure limit value in the magnitude of up to 100 mbar is exceeded and when a specific negative pressure limit value, that is not quantitatively above the overpressure limit value, between the pressure in the inner chamber of the tank and the surrounding pressure is fallen short of, and is otherwise closed, as well as a first storage unit for gaseous fuel components connecting to said valve unit; and • the connection between the compensation volume of the volume modifying element and the surroundings is guided through a second storage unit for gaseous fuel components; • wherein the first storage unit, when viewed from the inner chamber of the tank, connects to the valve unit; • wherein the second storage unit connects to the first storage unit in the direction of the surroundings; and • wherein a ventilation line of the compensation volume of the volume modifying element i) runs into a connection between the first storage unit, which is filled with an adsorbent for gaseous fuel components, and the second storage unit, ii) runs into the first storage unit, or iii) runs into a tank ventilation line, comprising the valve unit, upstream of the first storage unit and downstream of the valve unit. • the inner chamber of the fuel tank is connected to the surroundings via a valve unit, which is open when the fuel tank is being filled as well as when an overpressure limit value in the magnitude of up to 100 mbar is exceeded and when a specific negative pressure limit value, that is not quantitatively above the overpressure limit value, between the pressure in the inner chamber of the tank and the surrounding pressure is fallen short of, and is otherwise closed, as well as a first storage unit for gaseous fuel components connecting to said valve unit; and • the connection between the compensation volume of the volume modifying element and the surroundings is guided through a second storage unit for gaseous fuel components; • wherein the first storage unit, when viewed from the inner chamber of the tank, connects to the valve unit; • wherein the second storage unit connects to the first storage unit in the direction of the surroundings; and • wherein a ventilation line of the compensation volume of the volume modifying element i) runs into a connection between the first storage unit, which is filled with an adsorbent for gaseous fuel components, and the second storage unit, ii) runs into the first storage unit, or iii) runs into a tank ventilation line, comprising the valve unit, upstream of the first storage unit and downstream of the valve unit. i) runs into a connection between the first storage unit, which is filled with an adsorbent for gaseous fuel components, and the second storage unit, ii) runs into the first storage unit, or iii) runs into a tank ventilation line, comprising the valve unit, upstream of the first storage unit and downstream of the valve unit.