Question

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 498 cm3 of air at atmospheric pressure (1.01×105Pa) and a temperature of 27.0 ?C. At the end of the stroke, the air has been compressed to a volume of 46.4 cm3 and the gauge pressure has increased to 2.80×106 Pa . Compute the final temperature.

Answer 1

Construct two versions of the ideal gas law:
P_1*V₁ = n*R*T₁
P_2*V₂ = n*R*T₂

from equation 1
n*R = P_1*V₁/T₁

from equation 2
T₂ = P_2*V₂/(n*R)

Substitute and simplify:
T₂ = T_1*P_2*V₂/(P_1*V₁)
Absolute pressure at state 2 in terms of state 2 gauge pressure (P₂g):
P₂ = P₂g + background pressure
Our background pressure is identical to the pressure at intake state, thus:
P₂ = P₂g + P₁

Thus:
T₂ = T_1*(P₂g + P₁)*V₂/(P_1*V₁)

given data
T₁=300.15 K

P₂g=2800 kPa; P₁=101 kPa

V₂=46.4 cm³; V₁=498 cm³;

T₂ =300.15 K *(2800 KPa + 101 KPa)*46.4 cm³/(101 KPa*498 cm³)

T2 = 803.25 Kelvin

And convert into Celsius:
T2 = 530 Celsius