Answer
(a) $p_a = 12.3 atm$
(b) $Q_H = 5470.2 J$
(c) $Q_C = -3722.8 J$
(d) $W = 1747.4 J$
(e) $𝜖= 31.9 \%$
Work Step by Step
(a) The pressure of gas at point a is
$p_a 𝑉_a^𝛾= p_b V_b^𝛾$
$p_a = \frac{p_b V_b^𝛾}{𝑉_a^𝛾} $
$p_a = p_b (\frac{V_b}{𝑉_a})^𝛾 $
$p_a = (1.50 atm) (\frac{9.0 \times 10^{-3} m^3}{2.0 \times 10^{-3} m^3})^{1.4} $
$p_a = 12.3 atm$
(b) Heat enters the gas through process ca,
$Q_H = \frac{C_V V \Delta p}{R} $
$Q_H = \frac{(5/2 R) V \Delta p}{R} $
$Q_H = \frac{5 V \Delta p}{2} $
$Q_H = \frac{5 (2.0 \times 10^{-3} m^3 ) (12.3 atm - 1.5 atm) (1.013 \times 10^5 Pa/atm)}{2} $
$Q_H = 5470.2 J$
(c) The heat leaves through process bc.
$Q_C = \frac{C_p p \Delta V}{R} $
$Q_C = \frac{(7/2 R) p \Delta V}{R} $
$Q_C = \frac{7 p \Delta V}{2} $
$Q_C = \frac{7 (1.50 atm) (1.013 \times 10^5 Pa/atm) (2.0 \times 10^{-3} m^3 - 9.0 \times 10^{-3} m^3)}{2} $
$Q_C = -3722.8 J$
(d) Work the engine does
$W = Q_H + Q_C$
$W = 5470.2 J + (-3722.8 J) $
$W = 1747.4 J$
(e) $𝜖= \frac{𝑊}{|𝑄_𝐻 |} \times 100 $
$𝜖= \frac{1747.4 J}{5470.2 J} \times 100 $
$𝜖= 31.9 \%$
