Coefficient of performance of refrigerator

[Elias Waranoi]

Homework Statement

The pV-diagram in Fig. P20.51 (See attached file) shows the cycle for a refrigerator operating on 0.850 mol of H₂. Assume that the gas can be treated as ideal. Process ab is isothermal. Find the coefficient of performance of this refrigerator.

Homework Equations

K = Q_C/|Q_H - Q_C|
pV = nRT
C_p = C_v + R
Isothermal expansion: Q = W = ∫p dV = nRT*ln(V₂/V₁)
Constant volume: Q = nC_vΔT
Constant pressure: Q = nC_pΔT

The Attempt at a Solution

Q_C = Q_a-b + Q_b-c = nRT_a * ln(V_b/V_a) + nC_v(T_c - T_b) = p_aV_a * ln(V_b/V_a) + V_bC_v(p_c - p_aV_a/V_b)

Q_H = Q_c-a = n(C_v + R)(T_a - T_c) = p_c(C_v + R)(V_a - V_c)/R

My answer K = Q_C/|Q_H - Q_C| = 0.462
Correct answer K = 6.23

What did I do wrong?

 

[TSny]

Q_C = Q_a-b + Q_b-c = nRT_a * ln(V_b/V_a) + nC_v(T_c - T_b) = p_aV_a * ln(V_b/V_a) + V_bC_v(p_c - p_aV_a/V_b)

Did you leave out R somewhere in the last term on the right side?

Q_H = Q_c-a = n(C_v + R)(T_a - T_c) = p_c(C_v + R)(V_a - V_c)/R

Does this give a positive or negative value for Q_H? In the formula K = Q_C/|Q_H - Q_C|, is Q_H considered as positive or negative?

 

[Elias Waranoi]

Q_C = p_aV_a * ln(V_b/V_a) + V_bC_v(p_c - p_aV_a/V_b)/R

Q_H is heat the leaves the working substance so it should be negative. Apparently K = |Q_C|/( |Q_H| - |Q_C| ), thanks for the help I got the correct answer now.