Question

Gasoline vapor is injected into the cylinder of an automobile engine when the piston is in its expanded position. The temperature, pressure, and volume of the resulting gas-air mixture are 20 °C, 1.00x105N/m², and 240 cm3, respectively. The mixture is then compressed adiabatically to a volume of 40 cm3. (Note that in the actual operation of an automobile engine, the compression is not quasi-static, although we are making that assumption here). a) (10 pts) What are the pressure and temperature of the mixture after the compression? b) (10 pts) How much work is done by the mixture during the compression?

Answer 1

a) the pressure and temperature of the mixture after the compression are approximately 8.13x105 N/m² and 439 K, respectively.

b)the work done by the mixture is zero.

Step-by-step explanation:

we can use the ideal gas law and the adiabatic compression equation.

a) Using the ideal gas law, we can find the initial number of moles of gas-air mixture in the cylinder:

PV = nRT

n = PV/RT = (1.00x105 N/m²)(240 cm³/1000 cm³/m³)/(8.31 J/mol•K)(20+273 K) ≈ 0.027 mol

Since the mixture is compressed adiabatically, we can use the adiabatic compression equation to find the final pressure and temperature:

P1V1^γ = P2V2^γ

where γ = Cp/Cv is the ratio of specific heats of the mixture.

Assuming the gas-air mixture behaves as a diatomic gas, we can use the values γ = 1.4 and Cp = 29.1 J/mol•K and Cv = 20.8 J/mol•K.

Substituting the given values, we can solve for the final pressure and temperature:

P2 = P1(V1/V2)^γ = (1.00x105 N/m²)(240 cm³/40 cm³)^1.4 ≈ 8.13x105 N/m²

T2 = T1(V1/V2)^(γ-1) = (20+273 K)(240 cm³/40 cm³)^0.4 ≈ 439 K

Therefore, the pressure and temperature of the mixture after the compression are approximately 8.13x105 N/m² and 439 K, respectively.

b) The work done by the mixture during the compression can be found using the equation:

W = -ΔU

where ΔU is the change in internal energy of the mixture.

Since the compression is adiabatic, there is no heat transfer and ΔU = Q = 0.

Therefore, the work done by the mixture is zero.