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?
Read up on Boyle's Law. You clearly haven't studied this section of your text.
since PV is constant,
multiplying the pressure by 15/2 multiplies the volume by 2/15
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To find the factor by which the volume changes during the compression stroke of the engine, we can use the adiabatic equation for an ideal gas.
The adiabatic equation for an ideal gas is given by:
P1 * V1^γ = P2 * V2^γ
Where:
P1 and P2 are the initial and final pressures, respectively,
V1 and V2 are the initial and final volumes, respectively,
γ is the heat capacity ratio or adiabatic index for the gas.
For a diatomic ideal gas, such as air, the value of γ is approximately 1.4.
In this case, the initial pressure is 2 atm (P1 = 2 atm), and the final pressure is 15 atm (P2 = 15 atm). We are asked to find the factor by which the volume changes, so we need to solve for V2 / V1.
Rearranging the adiabatic equation, we get:
V2 / V1 = (P1 / P2)^(1/γ)
Now, let's substitute the given values into the equation:
V2 / V1 = (2 atm / 15 atm)^(1/1.4)
Calculating the right side of the equation, we have:
V2 / V1 = (0.1333)^(1/1.4)
By evaluating the expression, we find that V2 / V1 is approximately equal to 0.685.
Therefore, the volume changes by a factor of approximately 0.685 during the compression stroke of the engine.