The formula to calculate the final temperature during adiabatic compression is:
\[ T2 = T1 \cdot \left( \frac{P2}{P1} \right)^{\frac{\gamma - 1}{\gamma}} \]
Where:
Let's say the initial temperature (T1) is 300 K, the initial pressure (P1) is 100,000 Pa, the final pressure (P2) is 200,000 Pa, and the heat capacity ratio (γ) is 1.4. The final temperature would be calculated as follows:
\[ T2 = 300 \cdot \left( \frac{200000}{100000} \right)^{\frac{1.4 - 1}{1.4}} \approx 365.7 \text{ K} \]
So, the final temperature is approximately 365.7 K.
Adiabatic compression is a process in which the pressure of a gas is increased without any heat exchange with the surroundings. During this process, the temperature of the gas increases as work is done on it. This type of compression is commonly observed in various thermodynamic cycles, such as those in internal combustion engines and refrigeration systems. The relationship between pressure and temperature during adiabatic compression is governed by the heat capacity ratio (γ), which is the ratio of the specific heat at constant pressure to the specific heat at constant volume.