A diesel engine works at a high compression ratio to compress air until it reaches a temperature high enough to ignite the diesel fuel. Suppose the compression ratio (ratio of volumes) of a specific diesel engine is 19 to 1. If air enters a cylinder at 1 atm and is compressed adiabatically, the compressed air reaches a pressure of 66.0 atm. Assuming that the air enters the engine at room temperature (22.9°C) and that the air can be treated as an ideal gas, find the temperature (in K) of the compressed air.
To find the temperature of the compressed air, we can use the adiabatic compression equation:
P1 * V1^γ = P2 * V2^γ
Where:
P1 = initial pressure (1 atm)
V1 = initial volume
γ = adiabatic index for air (approximately 1.4, since air can be treated as an ideal gas)
P2 = final pressure (66.0 atm)
V2 = final volume
First, we need to find the initial volume, V1.
Since the compression ratio of the diesel engine is 19:1, we can calculate V1 using the formula:
V1 = V2 / (compression ratio)
Given that V2 is the final volume (which we don't know yet) and the compression ratio is 19:1, we can substitute these values:
V1 = V2 / 19
Now, rearrange the adiabatic compression equation to solve for V2:
V2^γ = P2 / P1 * V1^(-γ)
Now, substitute the known values into the equation:
V2^γ = (66.0 atm) / (1 atm) * (V2 / 19)^(-γ)
Now, let us convert the initial temperature from Celsius to Kelvin:
T1 = 22.9°C + 273.15 = 296.05 K
Because the air can be treated as an ideal gas, we can use the ideal gas law to calculate V1:
P1 * V1 = n * R * T1
Where:
n = number of moles of air
R = ideal gas constant (0.0821 L * atm / K * mol)
Since we don't know the number of moles, we can work with the relative values and solve for V1:
V1 = (P1 * V1) / (P1 * n * R * T1)
Now, we can substitute the known values into the equation:
V1 = (1 atm * V1) / (1 atm * n * R * 296.05 K)
Now, simplify the equation:
V1 = V1 / (n * R * 296.05)
Now, we have a relationship between V1 and V2, and can substitute the value for V1 into the adiabatic compression equation:
(V2 / 19)^γ = (66.0 atm) / (1 atm) * [V2 / (n * R * 296.05)]^(-γ)
Since we are only interested in finding the temperature (in Kelvin) of the compressed air, we can solve for V2:
(V2 / 19)^γ = (66.0 atm) * [(V2 / (n * R * 296.05)]^(-γ)
Raise both sides of the equation to the power of 1/γ:
V2 / 19 = (66.0 atm)^(1/γ) * [V2 / (n * R * 296.05)]^(-1)
Now, we can simplify the equation further:
V2 = 19 * (66.0 atm)^(1/γ) * [V2 / (n * R * 296.05)]^(-1)
Now, we have an equation with V2 on both sides, so let's isolate V2:
V2 = 19 * (66.0 atm)^(1/γ) / [1 + 19 * (66.0 atm)^(1/γ) / (n * R * 296.05)]
Finally, we need to convert the expression into an equation that gives us the temperature. We can use the ideal gas law again:
P2 * V2 = n * R * T2
Rearrange the equation to solve for T2:
T2 = (P2 * V2) / (n * R)
Now, substitute the known values into the equation:
T2 = (66.0 atm * V2) / (n * R)
Now, substitute the expression we derived for V2:
T2 = (66.0 atm * [19 * (66.0 atm)^(1/γ) / (1 + 19 * (66.0 atm)^(1/γ) / (n * R * 296.05))]) / (n * R)
Now, simplify the equation further:
T2 = (66.0 * 19 * (66.0)^(1/γ)) / (1 + 19 * (66.0)^(1/γ) / (n * R * 296.05))
Now, substitute the value of γ (approximately 1.4), R (0.0821 L * atm / K * mol), and solve for T2 using any suitable unit conversion, if required.