A gas undergoes a cycle in a piston-cylinder assembly consisting of the following three processes:

**Process 1-2:** Constant pressure, p = 1.4 bar, V_{1} = 0.028 m^{3}, W_{12} = 10.5 kJ

**Process 2-3:** Compression with pV = constant, U_{3 =} U_{2}

**Process 3-1:** Constant volume, U_{1} – U_{3} = -26.4 kJ

There are no significant changes in kinetic or potential energy.

(A) Sketch the cycle on a p-V diagram.

(B) Calculate the net work for the cycle, in kJ.

(C) Calculate the heat transfer for process 1-2, in kJ.

## Expert Answer

The first law of thermodynamics is a direct relation to the conservation of energy and states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but can neither be created nor destroyed.

The first law is formulated by stating that the change in the internal energy of a closed system is equal to the amount of heat supplied to the system minus the amount of work done by the system on its surroundings. Additionally, perpetual motion machines of the first kind are impossible.

In a constant pressure thermodynamic process, the pressure is always constant is always constant in the system. Constant pressure process is also called as isobaric process. This process is represented as a straight line on the p-V diagram.

In an isothermal thermodynamic process, the product of pressure and volume is always constant. For an ideal case, the change in internal energy during this process is zero. This process is represented as a rectangular hyperbola on the p-V diagram.

In a constant volume thermodynamic process, the volume is always constant is always constant in the system. Constant volume process is also called as isochoric process. Work done in constant volume process is zero.

Work done can be determined by finding the area under the pressure-volume diagram.

Write the formula for work done in constant pressure process.

Here, the pressure is and the change in volume is .

Write the equation of state.

Calculate the work done for a constant volume process.

Substitute for *p*.

In a process, the change in the internal energy of a system is equal to heat energy added to the system minus the work done by the system.

Taking as a change in internal energy, it is equal to,

…… (1)

Here, the heat supplied to the system from its surroundings is *Q* and the work done by the system is *W*.

**Sign Convention for work done**:** **Work done by the system is positive and work done on the system is negative.

**Sign convention for heat transfer**: Heat absorbed by the system is positive and heat rejection by the system is negative.

## Step-by-step

### Step 1 of 3

**a)**

Draw the *p *– *v *diagram of the system.

*p*–

*v*diagram of the given system is

Explanation | Hint for next step

Process 1-2 is a constant pressure process with workout from the system. So, on the p-V diagram it is represented with a horizontal line towards the right side.

Process 2-3 is a isothermal process . So, on the p-V diagram it is represented with a rectangular hyperbola.

Process 3-1 is a constant volume process with decrease in internal energy which indicates decrease in temperature. So, on the p-V diagram it is represented with a vertical line moving downward.

### Step 2 of 3

**b)**

Calculate the volume at state 2 using work done relation in constant pressure process.

Substitute for , for , and for .

Calculate work done in 2 -3, isothermal process using the following formula:

Substitute for , for , and for .

Negative sign indicates work is done on the system.

Work done in constant volume process (3-1) is zero.

Calculate the net work done in the cycle.

Substitute for , for , 0 for .

Explanation | Hint for next step

Work done in various processes in the cycle is calculated. The net work done in the cycle is sum of the work done in the individual processes.

### Step 3 of 3

**c)**

Calculate the heat transfer in 1-2 process using first law of thermodynamics relation.

Substitute for and for .

[Part C]

**Part C**

**Part C**

Explanation

Using first law of thermodynamics, the heat transfer in the process 1-2 is calculated.

### Answer

*p*–

*v*diagram of the given system is