A diesel engine's piston compresses 16 cm3 of fuel-air mixture into 1 cm3. The pressure changes from 1 atmosphere to 48 atmospheres. If the initial temperature of the gas was 305 K, what was the final temperature?
(P1V1/T1) = (P2V2/T2)
Don't forget T must be in kelvin.
1.09 x 10-3 K
To find the final temperature of the gas, we can use the ideal gas law equation: PV = nRT.
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
First, let's find the initial volume and final volume using the given information:
Initial volume (V1) = 16 cm^3
Final volume (V2) = 1 cm^3
Next, let's calculate the initial number of moles (n1):
Using the ideal gas equation, PV = nRT, we can rearrange it to solve for n1:
n1 = (PV1) / (RT1)
Substituting the values:
P = 1 atm
V1 = 16 cm^3
R = ideal gas constant (constant for all gases) = 0.0821 L.atm/K.mol
T1 = 305 K
n1 = (1 atm * 16 cm^3) / (0.0821 L.atm/K.mol * 305 K)
n1 = 0.00082035 mol
Next, let's calculate the final number of moles (n2):
Since the volume decreases from V1 to V2 and the gas remains constant (incompressible), the number of moles also remains constant, so n1 = n2.
Now, let's calculate the final temperature (T2):
Using the ideal gas equation, PV = nRT, we can rearrange it to solve for T2:
T2 = (PV2) / (nR)
Substituting the values:
P = 48 atm
V2 = 1 cm^3
n2 = 0.00082035 mol
R = 0.0821 L.atm/K.mol
T2 = (48 atm * 1 cm^3) / (0.00082035 mol * 0.0821 L.atm/K.mol)
T2 = 7120 K
Therefore, the final temperature of the gas is approximately 7120 K.