Question

The cylinder in an automobile engine has a radius of about 5 cm. The top of

the piston begins its compression stroke at about 23 cm from the head end

and travels about 18 cm during the stroke. The gas in the cylinder behaves

as an ideal gas with five degrees of freedom per molecule. It begins at 310 K

and 1 atm pressure when the piston is at the bottom of its stroke (i.e., fully

expanded).

(a) How many moles of gas are in the cylinder?

(b) What are the temperature and pressure of the gas at the end of the adia-

batic compression stroke?

(c) How much work is done on the gas by the piston during the compression

stroke?

(d) When the gas is completely compressed, the combustion of the gasoline

increases the temperature by another 800 K. What is the temperature

now?

(e) How much heat energy has been added to the gas?

f) How much work is done by the gas as it expands adiabatically back to

its starting point?

(g) What is the efficiency of this engine?

Answer 1

(a): Given T = 310 K, P = 1 atm

Volume of gas, Vi = r²h = 3.14 x (5 cm)²x23 cm = 1805.5 cm³ = 1.8055 L

Applying ideal gas equation:

PV = nRT

=> n = PV / RT = 1 atm x 1.8055 L / (0.0821 L.atm.mol-1.K-1 x 310 K)

=> moles of gas in the cylinder, n = 0.07094 mol or 0.071 mol (answer)

(b): Final volume after compression, V_f = = r²h = 3.14 x (5 cm)²x5 cm = 392.5 cm³ = 0.3925 L

Given degree of fredom, f = 5.

Hence = (f+2) / f = 7/5 = 1.4

For adiabatic compression, P_iV_i = P_fV_f  

=> P_f = P_i x (V_i/ V_f)

=> P_f = 1 atm x (1.8055L / 0.3925L)^1.4

=>  P_f = 8.47 atm (answer)

Also, for adiabatic compression, T_iV_i^-1 = T_fV_f^-1  

=> T_f = T_i x (V_i/ V_f)^-1

=> T_f = 310K x (1.8055L / 0.3925L)^1.4

=> T_f = 2626K (answer)

(c): The formula for the work done in adiabatic compression is

W = (PiVi - PfVf) / ( - 1)

=> W = (1 atmx1.8055L - 8.47 atmx0.3925L) / (1.4 - 1)

=> W = - 3.8 L.atm = - 3.8 x 10^-3m3 x 10⁵ M/m2 = - 3800 J (answer)

(d) T = T_f +800K = 2626K + 800K = 3426 K (answer)