4. The First Law of Thermodynamics for Closed Systems
4.3 Work
Work is a form of mechanical energy associated with a force and its resulting displacement. When a force

4.3.1 Boundary work
Work associated with the expansion and compression of a gas is commonly called boundary work because it is done at the boundary between a system and its surroundings.
Let us consider a piston-cylinder device, as illustrated in Figure 4.3.2. The gas in the cylinder exerts an upward force,
where
Specific boundary work refers to the boundary work done by a unit mass of a substance. It can be written as
where

From the integral equations for
Figure 4.3.3 demonstrates that the boundary work and specific boundary work in a quasi-equilibrium process are path functions; they depend on the initial and final states as well as the process path. Boundary work can be defined as positive or negative. Here is a common sign convention: the boundary work in an expansion process is positive. This is because the change of volume in an expansion process is positive. Likewise, the boundary work in a compression process is negative.

Example 1
Consider a rigid sealed tank of a volume of 0.3 m3 containing nitrogen at 10oC and 150 kPa. The tank is heated until the temperature of the nitrogen reaches 50oC. Treat nitrogen as an ideal gas.
- Sketch the process on a
diagram - Calculate the boundary work in this process
- Calculate the change in internal energy in this process
Solution
1.

2. The boundary work is zero because the volume of nitrogen remains constant in the process.
3. Change in internal energy in the process
From Table G1:
R=0.2968 kJ/kgK and Cv= 0.743 kJ/kgK
The mass of nitrogen:
The change in internal energy:
Nitrogen absorbs 15.9 kJ of heat in this process.
Example 2
Consider 0.2 kg of ammonia in a reciprocating compressor (piston-cylinder device) undergoing an isobaric expansion. The initial and final temperatures of the ammonia are 0oC and 30oC, respectively. The pressure remains 100 kPa in the process.
- Sketch the process on a
diagram - Calculate the boundary work in this process
- Calculate the change in internal energy in this process
Solution:
1.

2. Boundary work
From Table B2: for the initial state 1 at T = 0oC, P = 100 kPa,
For the final state 2 at T = 30oC, P = 100 kPa,
Graphically, the specific boundary work is the shaded rectangular area under the process line in the P –
3. Change in internal energy
Example 3
Consider air undergoing an isothermal expansion. The initial and final pressures of the air are 200 kPa and 100 kPa respectively. The temperature of the air remains 50oC in the process. Treat air as an ideal gas.
- Sketch the process on a
diagram - Calculate the specific boundary work in this process
- Calculate the change in specific internal energy in this process
Solution:
1.

2. Specific Boundary work
From Table G1: R = 0.287 kJ/kgK for air. The ideal gas, air, undergoes an isothermal process.
3. The process is isothermal; therefore, the temperature remains constant and the change in internal energy is zero.
4.3.2 Polytropic process and its boundary work
A polytropic process refers to any quasi-equilibrium thermodynamic process, which can be described with the following mathematical expression.
where
By adjusting
Process | Polytropic exponent | Ideal gas equation of state | Polytropic relation |
---|---|---|---|
Isobaric | |||
Isothermal | |||
Isochoric |
Figure 4.3.4 shows different polytropic processes of an ideal gas. In many actual thermodynamic processes, the polytropic exponents are typically in the range of

The boundary work and corresponding specific boundary work in a polytropic process can be calculated by using the following equations. Detailed derivations are left for the readers to practice.
The following expressions are valid for both real and ideal gases.
If
If
The following two expressions are valid only for ideal gases in an isothermal process (
If
where
Example 4
Consider an ideal gas undergoing a polytropic process. At the initial state: P1=200 kPa, v1=0.05 m3/kg. At the final state: v2=0.1 m3/kg. For n=1.3 and n=1,
- Sketch the two processes on a
diagram. Which process has a larger specific boundary work? - Calculate the specific boundary work and verify your answer to the question in part 1.
Solution:
1.
From the

2. For
For
Compare the specific boundary work in these two processes, the isothermal process (n=1) has a larger specific boundary work than the polytropic process with n = 1.3. The calculation results are consistent with the observation from the
4.3.3 Spring work
Spring work is a form of mechanical energy required to compress or expand a spring to a certain distance, see Figure 4.3.5. Spring force and spring work can be expressed as follows:
where

A linear spring with spring constant K=100 kN/m is mounted on a piston-cylinder device. At the initial state, the cylinder contains 0.15 m3 of gas at 100 kPa. The spring is uncompressed. The gas is then heated until its volume expands to 0.2 m3. The piston’s cross-sectional area is 0.1 m2. Assume the piston is frictionless with negligible weight and the process is quasi-equilibrium,
-
- Write an expression of the gas pressure as a function of the gas volume in this process
- Sketch the process on a
diagram - Calculate the total work done by the gas during this expansion process
- If the spring is not mounted on the piston, the gas in the cylinder will expand isobarically after being heated. To reach the same final volume, 0.2 m3, how much work must be done by the gas in the expansion process?
Figure 4.3.e5 Piston-cylinder device with a spring loaded on top of the piston
Solution:
1. Analyze the forces acting on the piston, see below.

Three forces acting on the piston are in equilibrium.
At the initial state:
with K =100 kN/m, A = 0.1 m2, Patm = 100 kPa and
where gas pressure is in kPa and
2.
From part 1, Pgas is a linear function of volume

3. During the expansion process, the gas has to overcome the resistance from the springand. At the same time, the gas pressure and volume increase until the gas reaches the final state. The total work done by the gas is the shaded area of the trapezoid in the
At the final state,
The total work done by the gas is
4. If the gas expands isobarically from

Practice Problems
Practice Problems
Media Attributions
- Boundary work © Kh1604 is licensed under a CC BY-SA (Attribution ShareAlike) license
- Work done by the spring force © Svjo is licensed under a CC BY-SA (Attribution ShareAlike) license
Boundary work refers to the work done by a substance at the system boundary due to the expansion or compression of the substance.
Specific boundary work is the boundary work done by one unit mass of a substance.