During the compression stroke of a certain gasoline engine, the pressure increases from 2atm to 15atm. If the process is adiabatic and the fuel–air mixture behaves as a diatomic ideal gas,
by what factor does the volume change?
PV is constant
if P changes by a factor of 15/2, V must change by a factor of 2/15
adiabatic P V^gamma = constant
The value gamma=1.4 is typical for any diatomic gas. Monatomic inert gases, on the other hand, such as Helium, Neon, and Argon, have gamma approx 1.6 . Carbon dioxide, which is triatomic, has a heat capacity ratio gamma=1.28
To determine the factor by which the volume changes during the compression stroke of the gasoline engine, we need to use the adiabatic compression formula for an ideal gas:
P1 * V1^γ = P2 * V2^γ
Where:
P1 = Initial pressure
V1 = Initial volume
P2 = Final pressure
V2 = Final volume
γ = Ratio of specific heats (for a diatomic gas, γ = 7/5)
In this case, we are given:
P1 = 2 atm
P2 = 15 atm
γ = 7/5
First, let's rearrange the equation:
V2^γ = (P1 * V1^γ) / P2
Now, we can substitute the given values:
V2^γ = (2 atm * V1^γ) / 15 atm
To get the factor by which the volume changes, we need to find V2/V1, which is equivalent to calculating the square root of V2^γ/V1^γ:
(V2 / V1) = (V2^γ / V1^γ)^(1/γ)
Plug in the values:
(V2 / V1) = ((2 atm * V1^γ) / 15 atm)^(1/γ)
Simplifying:
(V2 / V1) = (2/15)^(1/γ)
Substitute γ = 7/5:
(V2 / V1) = (2/15)^(5/7)
Calculating this expression will give you the factor by which the volume changes during the compression stroke of the gasoline engine.