To calculate the final pressure of the trapped air, we can use the ideal gas law equation:
PV = nRT
Where: P is the pressure V is the volume n is the number of moles R is the ideal gas constant T is the temperature in Kelvin
First, let's convert the temperatures from Celsius to Kelvin: Initial temperature (T1) = 27 °C = 27 + 273.15 K = 300.15 K Final temperature (T2) = 127 °C = 127 + 273.15 K = 400.15 K
Given: Initial volume (V1) = 4 m³ Initial pressure (P1) = 3 atm Final volume (V2) = V1/2 = 4 m³ / 2 = 2 m³
Now, we can calculate the initial number of moles (n1) using the ideal gas law:
n1 = (P1 * V1) / (R * T1)
Let's assume the air is an ideal gas and use the ideal gas constant: R = 0.0821 L·atm/(mol·K)
n1 = (3 atm * 4 m³) / (0.0821 L·atm/(mol·K) * 300.15 K) n1 = 0.591 moles
Since the number of moles remains constant, we can use the final number of moles (n2) to calculate the final pressure (P2) as follows:
n2 = n1
P2 = (n2 * R * T2) / V2
P2 = (0.591 moles * 0.0821 L·atm/(mol·K) * 400.15 K) / 2 m³ P2 ≈ 9.75 atm
Therefore, the final pressure of the trapped air, when compressed into half the volume and heated to 127 °C, is approximately 9.75 atm.