A gas is compressed adiabatically, which one of the following statement is NOT true?

We need to identify which statement is NOT true for adiabatic compression of a gas. An adiabatic process is one in which no heat is exchanged between the system and its surroundings, meaning $$Q = 0$$. The First Law of Thermodynamics states:
$$\Delta U = Q - W$$

For an adiabatic process ($$Q = 0$$):
$$\Delta U = -W$$

Option 1: "There is no heat supplied to the system." This is TRUE. By definition, an adiabatic process has $$Q = 0$$, meaning no heat enters or leaves the system.

Option 2: "There is no change in the internal energy." This is NOT TRUE. From the First Law with $$Q = 0$$, we have $$\Delta U = -W$$. During compression, work is done ON the gas, so the gas does negative work ($$W < 0$$). Therefore:
$$\Delta U = -W = -(\text{negative value}) > 0$$
The internal energy INCREASES during adiabatic compression. The statement $$\Delta U = 0$$ would only hold if $$W = 0$$ (no work done), which contradicts the fact that the gas is being compressed. Note: $$\Delta U = 0$$ is characteristic of an isothermal process for an ideal gas, not an adiabatic one.

Option 3: "The temperature of the gas increases." This is TRUE. Since $$\Delta U > 0$$, and for an ideal gas the internal energy depends only on temperature via $$U = nC_vT$$, an increase in internal energy means an increase in temperature. This is why adiabatic compression heats up a gas, as seen in diesel engines where adiabatic compression raises the air temperature enough to ignite fuel.

Option 4: "The change in internal energy is equal to the work done on the gas." This is TRUE. From $$\Delta U = -W$$, and noting that work done ON the gas is $$W_{on} = -W$$, we get $$\Delta U = W_{on}$$. Since no heat escapes ($$Q = 0$$), all the work done on the gas goes entirely into increasing its internal energy.

The statement that is NOT true is Option 2.
The correct answer is Option 2.