When supplying the heat supplied ($Q_H$), the temperature of the gas increases from $T_2$ to $T_3$ in an isochoric process (at constant volume). This implies that we can utilize the relationship for ERROR:8085 with the heat capacity at constant volume ($C_V$) and the variación de Temperature ($\Delta T$), expressed by the equation:

This results in the temperature in state 2 ($T_2$) and the temperature in state 3 ($T_3$) as follows:

(ID 15363)

By removing the absorbed heat ($Q_C$) when the volume ($V$) equals the compressed volume ($V_2$), the absolute temperature ($T$) increases from the temperature in state 1 ($T_1$) to the temperature in state 2 ($T_2$). This implies that we can use the relationship for ERROR:8085 with the heat capacity at constant volume ($C_V$) and the variación de Temperature ($\Delta T$), which is expressed by the equation:

this leads us to the expression:

(ID 15364)

The work is calculated using the integral of the work done on the system ($W_{in}$) with the number of moles ($n$) and the pressure ($p$), integrated in the volume ($V$), from the expanded volume ($V_1$) to the compressed volume ($V_2$):

If the pressure ($p$) is obtained using the universal gas constant ($R_C$), the number of moles ($n$), and the absolute temperature ($T$) with the gas equation

the integral for the absolute temperature ($T$) is equal to the temperature in state 1 ($T_1$).

$W = \displaystyle\int_{V_1}^{V_2} p dV = \displaystyle\int_{V_1}^{V_2} \displaystyle\frac{nRT_1}{V} dV = nRT_1\ln\left(\displaystyle\frac{V_2}{V_1}\right)$

Therefore,

(ID 15365)

The work is calculated using the integral of the work performed by the system ($W_{out}$) with the pressure ($p$), integrated in the volume ($V$), from the expanded volume ($V_1$) to the compressed volume ($V_2$):

If the pressure ($p$) is obtained using the universal gas constant ($R_C$), the number of moles ($n$), and the absolute temperature ($T$) with the gas equation

the integral for the absolute temperature ($T$) is equal to the temperature in state 1 ($T_1$).

$W = \displaystyle\int_{V_1}^{V_2} p dV = \displaystyle\int_{V_1}^{V_2} \displaystyle\frac{nRT_2}{V} dV = nRT_2\ln\left(\displaystyle\frac{V_2}{V_1}\right)$

Therefore,

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(ID 15756)

(ID 15757)

(ID 15758)

The efficiency ($\eta$) is defined as the ratio of the effective work ($W$) to the heat contributed to the system ($Q$):

where the effective work ($W$) is related to the work performed by the system ($W_{out}$) and the work done on the system ($W_{in}$) through:

while the heat contributed to the system ($Q$) is associated with the heat supplied ($Q_H$), which is defined as:

As the work performed by the system ($W_{out}$) is related to the number of moles ($n$), the temperature in state 2 ($T_2$), the expanded volume ($V_1$), the compressed volume ($V_2$), and the universal gas constant ($R_C$) through:

and the work done on the system ($W_{in}$) is associated with the temperature in state 1 ($T_1$) through:

and the heat supplied ($Q_H$) is linked to the heat capacity at constant volume ($C_V$) by:

the efficiency ($\eta$) can be calculated, resulting in:

(ID 15759)