To solve the problem of finding the resulting pressure when a gas is compressed adiabatically to half its original volume, we can use the adiabatic process equation. This equation relates the pressure and volume of a gas during an adiabatic process, which occurs without heat exchange with the environment. The relevant formula is:
Understanding the Adiabatic Process
For an ideal gas undergoing an adiabatic process, the relationship between pressure (P) and volume (V) can be expressed as:
P1 * V1^γ = P2 * V2^γ
Where:
- P1 = initial pressure
- V1 = initial volume
- P2 = final pressure
- V2 = final volume
- γ (gamma) = heat capacity ratio (given as 1.4 for air)
Given Values
From the problem, we know:
- Initial pressure, P1 = 1 atm (which we can convert to Pascals if needed, but we'll keep it in atm for simplicity)
- Initial volume, V1 = V
- Final volume, V2 = V/2 (since the volume is halved)
- γ = 1.4
Setting Up the Equation
Substituting the known values into the adiabatic equation:
1 atm * V^1.4 = P2 * (V/2)^1.4
Rearranging the Equation
We can rearrange this equation to solve for P2:
P2 = 1 atm * (V^1.4) / ((V/2)^1.4)
This simplifies to:
P2 = 1 atm * (V^1.4) / (V^1.4 / 2^1.4)
Notice that the V^1.4 terms cancel out:
P2 = 1 atm * 2^1.4
Calculating the Final Pressure
Now we need to calculate 2^1.4. Using a calculator, we find:
2^1.4 ≈ 2.639
Thus, substituting this value back into our equation gives:
P2 ≈ 1 atm * 2.639 ≈ 2.639 atm
Final Result
The resulting pressure after compressing the gas adiabatically to half its original volume is approximately 2.639 atm.
This demonstrates how the pressure increases significantly when the volume is reduced in an adiabatic process, reflecting the principles of thermodynamics in action. If you have any further questions or need clarification on any part of this process, feel free to ask!
