A possible ideal-gas cycle operates as follows. (i) from an...

A possible ideal-gas cycle operates as follows. (i) from an initial state $\left(p_1, V_1\right)$ the gas is cooled at constant pressure to $\left(p_1, V_2\right)$. (ii) the gas is heated at constant volume to $\left(p_2, V_2\right)$. (iii) the gas expands adiabatically back to $\left(p_1, V_1\right)$. Assuming constant heat capacities, show that the thermal efficiency is $$ 1-\gamma \frac{\left(V_1 / V_2\right)-1}{\left(p_2 / p_1\right)-1} $$

A possible ideal-gas cycle operates as follows.
(i) from an initial state $\left(p_1, V_1\right)$ the gas is cooled at constant pressure to $\left(p_1, V_2\right)$.
(ii) the gas is heated at constant volume to $\left(p_2, V_2\right)$.
(iii) the gas expands adiabatically back to $\left(p_1, V_1\right)$.
Assuming constant heat capacities, show that the thermal efficiency is
$$
1-\gamma \frac{\left(V_1 / V_2\right)-1}{\left(p_2 / p_1\right)-1}
$$

Key Concepts

Ideal Gas Cycle

This concept involves a series of thermodynamic processes applied to an ideal gas that bring the system back to its initial state, forming a cycle. Analysis of such cycles uses the first law of thermodynamics to relate heat exchanges and work done over multiple processes, providing insights into energy conversion and efficiency limits.

Isobaric Process

An isobaric process occurs at constant pressure. In such a process, changes in volume and temperature drive the energy exchange, and the heat added or removed is related to the change in enthalpy of the system. This principle is used to determine the heat transfer during cooling or heating at constant pressure.

Isochoric Process

An isochoric process, or constant-volume process, is characterized by no change in volume, meaning no work is done. Heat transfer in this process results solely in a change in the internal energy of the gas, which is directly proportional to the change in temperature. This aspect is central to calculating the input heat during the constant volume phase.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. For an ideal gas with constant heat capacities, the process follows the relation PV^? = constant, linking pressure, volume, and temperature. This relation is critical in connecting the states of the cycle and ensuring that the net work cycle can be determined.

Heat Capacities and Specific Heat Ratio (?)

Heat capacities at constant pressure (cp) and constant volume (cv) quantify the amount of energy required to raise the temperature of a gas by a unit amount under respective conditions. Their ratio, ? = cp/cv, is especially important in characterizing adiabatic processes and appears explicitly in the efficiency formula, influencing the cycle's performance.

Thermal Efficiency of a Cycle

Thermal efficiency measures the fraction of the heat input that is converted into net work during a cycle. It is determined by performing energy balances around each process in the cycle and comparing the net work output to the heat absorbed during the cycle. In context of this ideal gas cycle, the efficiency formula is derived by relating the process parameters from the isobaric, isochoric, and adiabatic processes.

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