Energy, Heat, Work and the First Law

Introduction and context

Thermodynamics is fundamentally concerned with energy: how it is stored, transferred, converted, and used. Mechanical engineers apply these ideas to engines, turbines, compressors, pumps, refrigeration systems, boilers, heat exchangers, batteries, and manufacturing processes. The first law of thermodynamics expresses the conservation of energy for thermodynamic systems. It does not state how rapidly a process occurs or whether it is possible in a particular direction; those questions belong mainly to thermodynamics’ second law and to transport phenomena.

A central distinction in this chapter is that energy is a property of a system, whereas heat and work are modes of energy transfer across a system boundary. Heat is transferred because of a temperature difference. Work is transferred when a generalized force acts through a displacement or when another organized energy-transfer mechanism operates. Neither heat nor work is stored in a system.

Learning objectives

By the end of this chapter, a student should be able to:

• distinguish total energy, internal energy, kinetic energy, and potential energy;
• distinguish heat and work from stored energy;
• calculate boundary, shaft, electrical, and other work;
• apply the first law to closed systems and thermodynamic cycles;
• select a suitable system boundary, sign convention, and set of assumptions;
• interpret energy balances for practical engineering equipment.

1. Energy as a property

A thermodynamic system is a specified quantity of matter or a region in space selected for analysis. Everything outside it is the surroundings. The system boundary may be real or imaginary, fixed or moving. A closed system contains a fixed amount of matter, although energy may cross its boundary. An open system, or control volume, allows mass and energy to cross its boundary; open systems are treated in a later chapter.

The total energy of a system is commonly written as

where E is total energy, U is internal energy, KE is kinetic energy, and PE is potential energy. For a system of mass m moving with velocity V at elevation z in a gravitational field g,

Here U has units of joules (J), V is in m/s, g is approximately 9.81 m/s² near Earth’s surface, and z is in m. The kinetic and potential terms are macroscopic forms of energy. Internal energy includes microscopic translational, rotational, vibrational, chemical, nuclear, and intermolecular contributions. In ordinary engineering thermodynamics, changes in nuclear energy are normally neglected.

Specific properties are properties per unit mass. Thus, specific internal energy is u = U/m, specific kinetic energy is V²/2, and specific potential energy is gz, all commonly expressed in kJ/kg. Extensive properties depend on system size, whereas intensive properties do not. Volume and total mass are extensive; pressure, temperature, and density are intensive.

Energy is a property and therefore a state function: its change depends only on the initial and final states. Heat and work are path functions: their values depend on the process path between those states. The differential notation reflects this distinction. An exact differential is written dE, while heat and work are conventionally written δQ and δW.

State, process, and equilibrium

A state is described by measurable properties such as pressure, temperature, and specific volume. A simple compressible substance at equilibrium is usually fixed by two independent intensive properties. A process is a change from one state to another. A cycle is a sequence of processes that returns the system to its initial state. For a cycle, the change in every property is zero, including total energy.

2. Heat and work as energy transfers

Heat transfer occurs when a temperature difference exists between a system and its surroundings. Under the sign convention used here, Q is positive when heat enters the system and negative when heat leaves it. Work transfer occurs when a force, electrical potential, shaft torque, or another generalized interaction transfers energy across the boundary. W is positive when work is done by the system on the surroundings and negative when work is done on the system.

With this convention, the first-law equation for a closed system is

or, in expanded form,

The symbols Q and W denote net energy transfers over a process, not properties possessed by the system. If an alternative convention defines work input as positive, the equation becomes ∆E = Q + W_in. Either convention is valid, but signs must be used consistently.

Heat transfer may occur by conduction, convection, radiation, or phase-change mechanisms. The first law does not require a particular heat-transfer mechanism; it only requires that the net heat crossing the boundary be accounted for. Likewise, work may occur through several mechanisms.

3. Work modes

3.1 Boundary or displacement work

For a piston–cylinder containing a gas, movement of the boundary produces displacement work. For a quasi-equilibrium process,

where p_ext is the pressure exerted by the surroundings on the moving boundary and V is system volume. The integral applies when the boundary moves through a succession of states that can be described mechanically. For a reversible process, p_ext differs infinitesimally from system pressure p, so

The units are Pa·m³ = J. If pressure is constant,

For expansion, V₂ > V₁ and boundary work is positive under the present convention. For compression, it is negative. A rigid tank has dV = 0 and therefore has no boundary work, even if its pressure changes.

The expression p∆V is valid only for constant external pressure. For a linear pressure–volume path, the work is the area under the path in a p–V diagram. Work is path-dependent: two processes joining the same end states can have different areas and different work values.

Worked example 1: constant-pressure expansion

A gas expands in a piston–cylinder from 0.10 m³ to 0.30 m³ against a constant external pressure of 200 kPa. Determine the boundary work.

Using the constant-pressure expression,

Since 1 kPa·m³ = 1 kJ, the gas does 40 kJ of work on the surroundings.

3.2 Shaft work

Shaft work is transmitted by a rotating shaft, as in turbines, compressors, pumps, mixers, and engine dynamometers. In differential form, mechanical shaft work can be represented as

where τ is torque in N·m and θ is angular displacement in radians. For constant torque,

For a rotating device operating at constant torque and angular speed ω, the power is

where Ẇ_sh is power in watts and ω is in rad/s. Since a radian is dimensionless, N·m/s = W. Actual devices may also have frictional losses, which must be included in an energy balance or efficiency definition.

3.3 Electrical work

When charge crosses a potential difference, electrical work is transferred. For a direct-current device with constant voltage and current,

where V_e is electrical potential difference in volts, I is current in amperes, and ∆t is time in seconds. The corresponding electrical power is

For time-varying voltage or current,

The sign depends on the chosen system boundary and current direction. An electric heater generally receives electrical work and transfers heat to the system contents. A generator may deliver electrical work to the surroundings while receiving shaft work.

3.4 Other work modes

Thermodynamic analyses may include several additional modes:

• Spring work: energy stored in or released from an elastic element.
• Surface-tension work: work associated with changing interfacial area.
• Magnetic and electric polarization work: relevant in specialized materials and devices.
• Gravitational work: often represented through the potential-energy term rather than as a separate boundary work mode.
• Stirring or paddle-wheel work: shaft work transferred through a mixer.

A useful modeling principle is to include a work mode explicitly only when it crosses the selected system boundary and is significant relative to other terms.

4. The first law for closed systems

The first law is the energy-conservation statement for a thermodynamic system. For a finite process,

where Q₁₂ and W₁₂ are net heat and work transfers from state 1 to state 2. Expanded,

If kinetic and potential-energy changes are negligible,

This reduced form is widely used for stationary piston–cylinder systems, rigid tanks, and many laboratory devices.

For a differential process,

The equation is a balance, not a constitutive relation. To evaluate it, the analyst must obtain additional information about the substance, process path, heat transfer, or work interaction. The first law alone does not determine temperature, pressure, or heat transfer without such information.

Special cases

1. Adiabatic process: Q = 0. Therefore ∆E = −W. If the system does work without heat input, its stored energy decreases.
2. Rigid closed system: ∆V = 0, so boundary work is zero. Other work modes may still occur.
3. No work interaction: W = 0, so ∆E = Q.
4. Steady state of a closed system: A truly closed system cannot continuously operate at steady state unless its stored energy remains constant; transient changes are usually more relevant.
5. Cycle: E₂ = E₁, so

The net heat supplied during a cycle equals the net work delivered. A power cycle has net work output; a refrigeration or heat-pump cycle requires net work input.

Worked example 2: electrical heating of a rigid tank

A rigid, well-insulated tank contains a substance whose internal energy increases by 900 kJ. An electrical heater supplies 1,000 kJ to the system. Determine the paddle-wheel work done by the system, assuming changes in kinetic and potential energy are negligible.

For a rigid tank, boundary work is zero. The first law gives

The tank is well insulated, so Q = 0. Thus,

Therefore W = −900 kJ. The negative sign means 900 kJ of net work is done on the system. The electrical input is 1,000 kJ, so the remaining 100 kJ must be associated with work delivered to the system through the paddle wheel only if the stated internal-energy increase and heat assumption are interpreted consistently as separate transfers. More directly, if the heater is the only energy input and no shaft work occurs, the expected internal-energy increase would be 1,000 kJ. Hence the data imply an additional 100 kJ of work output, not paddle-wheel work input. This illustrates why every energy interaction and sign must be specified clearly.

A clearer equivalent statement is: if Q = 0 and ∆U = 900 kJ, then net work is −900 kJ. If 1,000 kJ of electrical work enters and no other interaction exists, the data are inconsistent unless 100 kJ leaves through another work mode. Energy balances are valuable for detecting such inconsistencies.

5. Energy accounting and reference states

Internal energy is often known only relative to a selected reference state. This causes no difficulty because engineering balances use differences such as u₂ − u₁. Absolute values are unnecessary unless a property model provides them. For ideal gases, internal energy is primarily a function of temperature, so

If the specific heat at constant volume is approximately constant,

where c_v is in kJ/(kg·K), temperatures may be expressed in kelvins or degrees Celsius for a temperature difference, and ∆u is in kJ/kg. For mass m, ∆U = m∆u. The constant-specific-heat approximation should be checked against the temperature range and available property data.

For a solid or liquid in a moderate temperature range, a similar approximation may be used with an appropriate specific heat. For phase changes, property tables or suitable equations of state are required because latent-energy effects can be large.

6. Thermodynamic cycles

A cycle returns a system to its initial state. Although ∆U, ∆KE, and ∆PE are zero over a complete cycle, heat and work transfers need not be zero. For a power cycle,

A thermal efficiency is defined as

where Q_in is the total heat supplied during the cycle. For a refrigerator, the desired effect is heat removal from a low-temperature region. Its coefficient of performance is

where Q_L is heat removed from the low-temperature space. For a heat pump,

where Q_H is heat delivered to the high-temperature region. These performance measures are not efficiencies in the strict sense and can exceed unity because they measure heat moved per unit work, not conversion of heat into work.

On a pressure–volume diagram, the net work of a quasi-equilibrium cycle is the signed area enclosed by the cycle. Clockwise cycles generally represent net work output; counterclockwise cycles generally represent net work input. This graphical interpretation is useful for engines, compressors, and refrigeration cycles.

7. A systematic solution method

A reliable first-law analysis follows these steps:

1. Define the system. State whether it is closed or open and draw an imaginary boundary.
2. Identify the process. Determine whether the boundary moves, whether mass is fixed, and whether the process is cyclic or transient.
3. Choose a sign convention. Here, heat into and work out of the system are positive.
4. List energy interactions. Include heat, boundary work, shaft work, electrical work, and changes in kinetic and potential energy.
5. Write the general balance first. Use ∆U + ∆KE + ∆PE = Q − W before making approximations.
6. Apply justified assumptions. For example, rigid means ∆V = 0; insulated means Q ≈ 0; stationary means ∆KE ≈ 0.
7. Use property data. Obtain internal energy, enthalpy, or specific heat from appropriate tables, charts, or validated models.
8. Check units and signs. Confirm that all energy terms use J or kJ and interpret the sign physically.
9. Assess plausibility. Compare the result with expected magnitudes and conservation of energy.

8. Applied case: energy balance in a laboratory air compressor

Background. An engineering laboratory evaluates a small electrically driven air compressor. Air enters the compressor continuously, is compressed, and leaves at a higher pressure and temperature. The motor receives electrical power, while the compressor rejects some heat to the surroundings.

Problem. The laboratory team initially treats the compressor as a closed system and attempts to use ∆U = Q − W. The calculated result predicts an impossible temperature decrease during compression.

Analysis. The compressor is not a closed system because air crosses the boundary. It must be modeled as a control volume. The electrical motor and compressor shaft may be included as one combined control volume or treated as separate devices. The correct analysis must account for mass-flow energy, shaft work, heat transfer, and changes in kinetic and potential energy. For steady operation, the stored energy of the control volume does not change, but energy carried by the flowing air does.

For a steady-flow device, the appropriate balance is conceptually

where Q̇ is heat-transfer rate into the control volume, Ẇ is work rate out of the control volume, ṁ is mass-flow rate, h is specific enthalpy, V is velocity, and z is elevation. The enthalpy h = u + pv combines internal energy with flow work. For a compressor, shaft work is usually input, so Ẇ is negative under the present convention.

Approach and solution. The team redraws the boundary around the compressor, measures inlet and outlet pressures and temperatures, estimates heat loss, and uses validated air-property data. It also separates motor electrical input from mechanical shaft output using measured efficiency. Kinetic and potential terms are retained initially and then shown to be small relative to the enthalpy and shaft-work terms.

Outcome. The revised model predicts an outlet temperature increase consistent with compression and measured data. The original error was not arithmetic; it was an incorrect choice of system boundary.

Learning points. System definition is often more important than algebra. Closed-system balances cannot be applied directly to steady-flow equipment. Measured electrical power is not automatically equal to thermodynamic work delivered to the fluid because motor and mechanical losses intervene.

9. Limitations, safety, and contemporary practice

The first law provides no information about energy quality, irreversibility, or the maximum possible efficiency. A process may satisfy energy conservation but still be impossible because it violates the second law. In real equipment, friction, turbulence, leakage, electrical resistance, heat loss, and nonuniform properties make idealized models approximate.

Engineers should state assumptions transparently, retain units, and avoid reporting excessive precision unsupported by measurements or property data. Energy balances also support safety analysis: unexpected heat accumulation can cause pressure rise, material failure, or thermal runaway. In industrial practice, instrumentation and digital twins increasingly use automated data acquisition and AI-assisted anomaly detection. Such tools can identify deviations from a calibrated energy balance, but they do not replace conservation laws, sensor validation, uncertainty analysis, or professional responsibility.

Chapter Summary

Energy is a property of a system and may be stored as internal, kinetic, or potential energy. Heat and work are not stored; they are energy-transfer modes crossing a system boundary. Boundary work is the integral of external pressure with respect to volume, shaft work is associated with torque and angular displacement, and electrical work is associated with voltage and charge transfer. With heat into and work out of the system taken as positive, the first law for a closed system is ∆E = Q − W. Special forms apply to adiabatic, rigid, stationary, and cyclic processes. Over a cycle, net heat equals net work. Correct system selection, sign conventions, property data, unit consistency, and justified assumptions are essential for reliable engineering analysis. The first law conserves energy but does not determine energy quality or process feasibility.

Key Terms

Energy: The capacity for transfer or transformation, measured in joules.

Internal energy: Microscopic energy associated with molecular and atomic structure and motion.

Kinetic energy: Energy associated with macroscopic motion, mV²/2.

Potential energy: Energy associated with elevation in a gravitational field, mgz.

Heat: Energy transfer caused solely by a temperature difference.

Work: Organized energy transfer associated with a generalized force acting through a displacement or equivalent interaction.

Boundary work: Work caused by movement of a system boundary.

Shaft work: Work transmitted by a rotating shaft.

Electrical work: Energy transfer associated with electric potential and charge flow.

Closed system: A system with fixed mass across whose boundary energy may pass.

Open system: A control volume through which mass and energy may pass.

State function: A property whose change depends only on the end states.

Path function: A quantity whose value depends on the process path.

Process: A change from one thermodynamic state to another.

Cycle: A sequence of processes returning the system to its initial state.

Adiabatic process: A process with negligible heat transfer.

Rigid system: A system with constant volume and no boundary work.

First law: The conservation-of-energy principle applied to thermodynamic systems.

Thermal efficiency: Net work output divided by heat supplied in a power cycle.

Coefficient of performance: Desired refrigeration or heating effect divided by work input.

Review Questions

Short Answer Questions

1. Why are heat and work classified as path functions rather than properties?
2. State the sign convention used for Q and W in this chapter.
3. Under what condition is boundary work zero?
4. Distinguish intensive and extensive properties, giving one example of each.
5. What does the first law imply for the net heat and work of a complete cycle?

Long Answer Questions

1. Explain the distinction between internal, kinetic, and potential energy, and discuss when each term may be neglected.
2. Derive the constant-pressure boundary-work expression from the general work integral and state its assumptions.
3. Compare boundary, shaft, and electrical work, including one engineering device associated with each mode.
4. Explain how the first law is applied to an adiabatic rigid tank and identify the possible remaining energy interactions.
5. Discuss why system-boundary selection is essential when analyzing compressors, turbines, and electric motors.

Analytical / Application Questions

1. A piston–cylinder expands from 0.15 m³ to 0.50 m³ against a constant external pressure of 150 kPa while receiving 80 kJ of heat. Neglect kinetic and potential-energy changes. Determine the boundary work and change in internal energy.
2. A rotating shaft transmits a constant torque of 25 N·m through 120 revolutions. Determine the shaft work in joules and explain its sign if the shaft drives a compressor.
3. A rigid, insulated vessel receives 2.4 MJ of electrical energy and delivers 0.4 MJ of shaft work to an agitator. Determine the change in internal energy, neglecting kinetic and potential-energy changes, and explain the physical meaning of the result.

Multiple Choice Questions with Answers

Question 1

Which quantity is a property of a thermodynamic system?

A. Heat transfer

B. Boundary work

C. Internal energy

D. Shaft work

Correct Answer: C. Internal energy

Question 2

A closed system follows the sign convention that heat entering is positive and work done by the system is positive. Its first-law equation is:

A. ∆E = Q + W

B. ∆E = Q − W

C. ∆E = W − Q

D. ∆E = −Q − W

Correct Answer: B. ∆E = Q − W

Question 3

A gas expands from 0.10 m³ to 0.25 m³ against a constant pressure of 300 kPa. What is the boundary work?

A. 20 kJ

B. 30 kJ

C. 45 kJ

D. 75 kJ

Correct Answer: C. 45 kJ

Question 4

Which condition guarantees zero boundary work for a closed system?

A. Constant temperature

B. Constant pressure

C. Constant volume

D. Zero heat transfer

Correct Answer: C. Constant volume

Question 5

For a shaft rotating at constant torque τ and angular speed ω, the shaft-power magnitude is:

A. τ/ω

B. τω

C. ω/τ

D. τ + ω

Correct Answer: B. τω

Question 6

During a complete thermodynamic cycle, the change in internal energy is:

A. Equal to the heat supplied

B. Equal to the work output

C. Zero

D. Always positive

Correct Answer: C. Zero

Question 7

A rigid, insulated tank receives 500 kJ of electrical work and has no other energy interaction. Neglecting kinetic and potential energies, its internal energy change is:

A. −500 kJ

B. 0 kJ

C. +250 kJ

D. +500 kJ

Correct Answer: D. +500 kJ

Question 8

Which statement about heat and work is correct?

A. Both are stored properties

B. Heat is a property but work is not

C. Both are modes of energy transfer

D. Work can cross a boundary only when temperature differs

Correct Answer: C. Both are modes of energy transfer

Question 9

A compressor operating continuously with mass entering and leaving should normally be analyzed as a:

A. Closed system

B. Control volume

C. Rigid system only

D. Thermodynamic cycle only

Correct Answer: B. Control volume

Question 10

A power cycle receives 1,000 kJ of heat and produces 300 kJ of net work. Its thermal efficiency is:

A. 0.30

B. 0.70

C. 1.30

D. 3.33

Correct Answer: A. 0.30