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?
PV=kT
So, PV/T = k a constant.
So, you want T such that
48*1/T = 1*16/305
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 = gas constant
T = temperature
Given:
Initial volume (V1) = 16 cm^3
Final volume (V2) = 1 cm^3
Initial pressure (P1) = 1 atmosphere
Final pressure (P2) = 48 atmospheres
Initial temperature (T1) = 305 K
First, let's find the number of moles (n) using the initial volume and pressure.
Using the ideal gas law formula, rearrange to solve for n:
n = (P1 * V1) / (R * T1)
Substituting the given values:
n = (1 atm * 16 cm^3) / (R * 305 K)
Since we are not given the gas constant (R), we can use the most common value for R, which is 0.0821 L*atm / (mol*K).
So, n = 0.000513 mol
Now that we have the number of moles, we can use it to find the final temperature.
Using the new volume and pressure, rearrange the ideal gas law equation again to solve for T2:
T2 = (P2 * V2) / (n * R)
Substituting the values:
T2 = (48 atm * 1 cm^3) / (0.000513 mol * 0.0821 L*atm/(mol*K))
T2 ≈ 23529 K
Therefore, the final temperature of the gas in the diesel engine is approximately 23529 K.