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?
Note: As long as the units for pressure and volume are the same on both sides of the equation, they will cancel. Temperature, however, must be in units of kelvin. Be sure to use the proper number of significant figures.
P2V2/T2=P1V1/T1
solve for T2
T2=P1V1/(T1P2V2)
temps in kelvins.
To find the final temperature, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure (in atmospheres)
V = volume (in cm^3)
n = number of moles of gas (which we can assume to be constant)
R = ideal gas constant (0.0821 L*atm/mol*K)
T = temperature (in Kelvin)
First, let's convert the initial and final volumes from cm^3 to liters by dividing by 1000:
Initial volume (V1) = 16 cm^3 / 1000 = 0.016 L
Final volume (V2) = 1 cm^3 / 1000 = 0.001 L
Now, let's rearrange the ideal gas law equation to solve for the final temperature (T2):
T2 = (P1 * V1 * T1) / (P2 * V2)
Plugging in the known values:
P1 = 1 atm
T1 = 305 K
P2 = 48 atm
V1 = 0.016 L
V2 = 0.001 L
R = 0.0821 L*atm/mol*K
T2 = (1 atm * 0.016 L * 305 K) / (48 atm * 0.001 L)
T2 = (0.016) * (305) / (48)
T2 = 0.16 K
So, the final temperature of the gas is approximately 0.16 K.
To find the final temperature of the gas, we can use the ideal gas law equation, which is given by:
PV = nRT
Where:
P is the pressure
V is the volume
n is the number of moles of gas
R is the ideal gas constant (0.0821 L·atm/mol·K)
T is the temperature in Kelvin
In this case, we are given:
P1 = 1 atm (initial pressure)
P2 = 48 atm (final pressure)
V1 = 16 cm^3 (initial volume)
V2 = 1 cm^3 (final volume)
T1 = 305 K (initial temperature)
First, we need to find the number of moles of gas (n) using the initial conditions. The moles of gas can be calculated using the following equation:
n = PV / RT
Substituting the given values:
n1 = (1 atm * 16 cm^3) / (0.0821 L·atm/mol·K * 305 K)
Next, we need to find the final temperature (T2) using the final conditions. Rearranging the ideal gas law equation, we can solve for T2:
T2 = (P2 * V2) / (n1 * R)
Substituting the given values:
T2 = (48 atm * 1 cm^3) / (n1 * 0.0821 L·atm/mol·K)
Now, let's plug in the values and complete the calculations:
n1 = (1 atm * 16 cm^3) / (0.0821 L·atm/mol·K * 305 K) ≈ 0.000694 moles
T2 = (48 atm * 1 cm^3) / (0.000694 moles * 0.0821 L·atm/mol·K) ≈ 873 K
Therefore, the final temperature of the gas is approximately 873 Kelvin.