Question

For an Otto cycle with compression ratio $r$ and specific heat ratio $\gamma$, which expression represents the thermal efficiency?

• A. $1 - \dfrac{1}{r^{\gamma-1}}$
• B. $1 - \dfrac{T_3 - T_4}{T_2 - T_1}$
• C. $\dfrac{(P_3 v_3 - P_4 v_4) - (P_2 v_2 - P_1 v_1)}{C_v (T_3 - T_2)(\gamma - 1)}$
• D. $1 - r^{\gamma - 1}$

Hint:

For the Otto cycle, efficiency depends only on compression ratio and $\gamma$: $\eta = 1 - 1/r^{\gamma-1}$.

Answer 1

The Correct Option is A, C

The thermal efficiency of an ideal Otto cycle is:
\[ \eta = 1 - \frac{1}{r^{\gamma - 1}} \]
Therefore, option (A) is correct.
Option (D) is incorrect because the expression is inverted and becomes negative for $r>1$.
Option (B) is not the correct general expression for an Otto cycle; it does not match the thermodynamic efficiency relation.
Option (C) represents the thermal efficiency using work done over heat input, expressed in $P v$ and $T$ terms. This is a valid expanded thermodynamic form consistent with the Otto cycle relations, so (C) is correct.
Thus, the correct answers are A and C.