You are correct that Charles's Law states that temperature and volume are directly proportional for a fixed amount of gas at constant pressure. However, it is important to note that Charles's Law assumes that other variables, such as the amount of gas and pressure, remain constant.
When you compress air in a syringe to reduce its volume, you are increasing the pressure while keeping the amount of gas constant. As a result, the temperature of the compressed air will also increase due to the increased pressure.
Charles's Law describes the relationship between temperature and volume when pressure and the amount of gas are held constant. It does not account for changes in pressure or the resulting increase in temperature when compressing or expanding gases.
In your example, as you compress the air in the syringe, the pressure increases, and according to the ideal gas law (PV = nRT), the temperature will also increase to maintain the relationship. However, it does not mean that the air will reach 300°C simply by compressing it to half its volume.
The increase in temperature due to compression is a result of the energy associated with the increased pressure. It is not solely determined by the initial temperature and the change in volume. To accurately determine the final temperature, you would need to consider additional factors such as the specific gas being compressed, the heat transfer during compression, and any heat loss or gain from the surroundings.
In summary, Charles's Law is applicable when the pressure and the amount of gas remain constant, but when these factors change, as in the case of compressing air in a syringe, the relationship between temperature and volume becomes more complex, and additional considerations are required to determine the final temperature.