During the compression stroke of a certain gasoline engine, the pressure increases from 1.00atm to 20.0atm.
If the process is adiabatic and the fuel air mixture behaves a a diatomic ideal gas
a) by what factor does the volume change
b) by what factor does the temperature change?
c) assumignt he compression starts with 0.0160mol of gas at 27.0 C, find the values of Q,W, and Eint that characterize the process
P*V^(7/5) = constant for adiabatic compression of a diatomic gas.
(P2/P1) = (V2/V1)^-7/5 = 20
The volume ratio is (20)^(-5/7)
Once you have the P ratio and the V ratio, use the gas law to get the T ratio
PV/T = constant
P2 V2/T2 = P1 V1/T1
T2/T1 = (P1 V1)/(P2 V2)
Thank You drwls
To answer these questions, we can use the adiabatic equation for an ideal gas:
\(PV^\gamma = \text{constant}\)
Where:
P = pressure
V = volume
\(γ\) = specific heat ratio
For diatomic ideal gases, the specific heat ratio \(γ\) is equal to 1.4.
Now let's calculate the answers to each question:
a) By what factor does the volume change?
Since we have an adiabatic process, we can use the adiabatic equation:
\(P_1V_1^\gamma = P_2V_2^\gamma\)
Where the subscripts 1 and 2 denote the initial and final states, respectively.
We know that the initial pressure, \(P_1\), is 1.00 atm and the final pressure, \(P_2\), is 20.0 atm.
Taking the ratio of the initial and final volumes and isolating the volume ratio, we get:
\(V_2 / V_1 = (P_1 / P_2)^{1/\gamma}\)
Substituting the given values, we have:
\(V_2 / V_1 = (1.00 \text{ atm} / 20.0 \text{ atm})^{1/1.4}\)
Calculating this expression gives us the factor by which the volume changes.
b) By what factor does the temperature change?
The temperature change can be calculated using the ideal gas law:
\(PV = nRT\)
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
Since we have an adiabatic process, the change in temperature can be calculated using the adiabatic equation:
\(T_2 / T_1 = (P_1 / P_2)^{(\gamma - 1)}\)
Here, \(T_1\) is the initial temperature and \(T_2\) is the final temperature.
Using the same values for \(P_1\) and \(P_2\) as in part a), we can calculate the factor by which the temperature changes.
c) Assuming the compression starts with 0.0160 mol of gas at 27.0 °C, find the values of Q, W, and \(E_{int}\) that characterize the process.
To find the values of heat transfer (Q), work (W), and internal energy change (\(E_{int}\)), we can use the first law of thermodynamics:
\(\Delta E_{int} = Q - W\)
In an adiabatic process, there is no heat transfer (\(Q = 0\)) because the process is thermally insulated.
Therefore, \(\Delta E_{int} = -W\).
Since we don't have the value for \(W\), we need to find it using the equation:
\(W = \frac{γ}{γ - 1} (P_2V_2 - P_1V_1)\)
Substituting the given values for \(P_1\), \(P_2\), \(V_1\), and the volume ratio from part a), we can find the value of \(W\).
Finally, using \(\Delta E_{int} = -W\) and the ideal gas law, we can calculate the change in internal energy (\(\Delta E_{int}\)).
By following these steps, we can answer all the questions and find the values of the required parameters.