One cylinder in the diesel engine of a truck has an initial volume of 650 cm3. Air is admitted to the cylinder at 35 ∘C and a pressure of 1.0 atm. The piston rod then does 450 J of work to rapidly compress the air.

A. What is its final temperature?

Express your answer to two significant figures and include the appropriate units

B.

What is its final volume?

Express your answer to two significant figures and include the appropriate units.

1) the final temperature is T2 = 876.76°C

2) the final volume is V2 = 24.14 cm³

Explanation:

We can model the gas behaviour as an ideal gas, then

P*V=n*R*T

since the gas is rapidly compressed and the thermal conductivity of a gas is low a we can assume that there is an insignificant heat transfer in that time, therefore for adiabatic conditions:

P*V^k = constant = C, k = adiabatic coefficient for air = 1.4

then the work will be

W = ∫ P dV = ∫ C*V^ (-k) dV = C*[ ((V2^ (-k+1) - V1^ (-k+1) ] / (-k + 1) = (P2*V2 - P1*V1) / (1-k) = nR (T2-T1) / (1-k) = (P1*V1/T1) * (T2-T1) / (1-k)

W = (P1*V1/T1) * (T2-T1) / (1-k)

T2 = (1-k) W * T1 / (P1*V1) + T1

replacing values (W=-450 J since it is the work done by the gas to the piston)

T2 = (1-1.4) * (-450J) * 308K / (101325 Pa*650*10^-6 m³) + 308 K = 1149.76 K = 876.76°C

the final volume is

TV^ (k-1) = constant

therefore

T2/T1 = (V2/V1) ^ (1-k)

V2 = V1 * (T2/T1) ^ (1 / (1-k)) = 650 cm³ * (1149.76K/308K) ^ (1 / (1-1.4)) = 24.14 cm³