This notes sheet looks at energy, heat, work & the first law of thermodynamics for processes and cycles.

In mechanics, the key forms of energy are kinetic and potential. These energies apply to a body as a whole, and so can be seen as macroscopic.

Microscopic energies are the forms of energy within that body: bond energies, kinetic and potential energies of individual particles (sometimes known as the ‘sensible’ energy), nuclear energy etc. All these microscopic energies are grouped together as internal energy, U.

The total energy of a system, E, is given as the sum of the internal, kinetic, and potential energies:


E = \frac{1}{2} mv^2 + mgz

By dividing by the mass, the specific energies can be found:

e = u + \frac{1}{2} v^2 + gz

Closed systems generally remain motionless, so rarely experience a change in kinetic or potential energies. In this case, the change in total energy is equal to the change in internal energy.

For control volumes, fluid may be moving in and out at a mass flow rate (represented by an m with a dot above it). In this instance, the energy flow rate is:

\dot{E} = \dot{m} e

We can group internal energies into two parts: the static energies and the dynamic energies. The former are fixed within the system, and so cannot change the overall internal energy. The latter are free to cross the system boundary, and so it is these that we are interested in.

There are only two forms of energy transfer for a closed system: heat transfer and work.

In a closed system, an energy transfer is always a heat transfer if it is brought about by a temperature difference. In all other situations, the energy transfer takes the form of a work transfer.

Heat Transfer

Heat transfer, Q, is the transfer of energy due to a difference in temperature between a system and its surroundings. In thermodynamics, the term ‘heat’ should only be used to refer to heat transfer. If you are talking about the energy due to temperature of a substance or system, call this ‘thermal energy’.

Heat transfer, heating up. coling down, thermodynamics, first law of thermodynamics

It is convention to define heat transfer from the surroundings to the system as positive, and heat transfer from the system to the surroundings as negative.

Heat transfer can take one of three forms:

  1. Conduction between neighbouring particles
  2. Convection between a solid and a fluid
  3. Radiation through electromagnetic waves or photons

Work Transfer

Work transfer, W, is the transfer of energy between a system and its surroundings that is not caused by a difference in temperature. Sometimes it is useful to think about work transfer as the lifting of a mass: if the energy transfer can in any way be modelled as raising a mass, then it is a work transfer and not a heat transfer.

The sign convention for work transfer is the opposite to that of heat transfer.

  • A positive work transfer means work transfer from the system to the surroundings
  • A negative work transfer means work transfer from the surroundings to the system

There are two key types of mechanical work transfer: displacement work and shaft work.

Displacement Work

Displacement work is the work done in moving a system boundary by a certain distance in the direction of a normal force. A common example is a piston moving due to a change in pressure, volume, and/or temperature. In this instance, the change in work is given as the product of the pressure and the change in volume:

\Delta W = P dV

If the expansion is fully resisted, it can be modelled as a quasi-equilibrium process. When this is the case, we can plot it on a P-V graph where the work is the area under the graph:

Work transfer, work done, calculating work done, P-V diagram, work done graph, thermodynamics

Therefore, the work done is given as the integral of P dV:

W = \int^{V_2}_{V_1} P dV

There are a few key forms that are useful to know, to spare integrating them every time:

  • Constant pressure:

W = P(V_2-V_1)

  • Constant temperature (PV is constant):

W = c \ln{ \frac{V_2}{V_1}}

  • Polytropic systems (PV^{\gamma} is constant):

W = \frac{1}{1-\gamma} (P_2 V_2 - P_1 V_1)

Shaft Work

Shear work is the work done in moving a system boundary in the direction of a tangential force (a shear force). This is very common in liquids, where they interact with their container. Shear work transfer into a fluid typically comes from a stirring force:

Shaft work, thermodynamics, rotary work done, shaft work done

Instead of modelling the system as just the fluid, we can model the system as the fluid, the paddle, and the shaft. This means the only input to the system is a shear work on the shaft, known as shaft work.

The force involved is a torque couple, and so the power of the shaft (the rate of work transfer) is given as:

Shaft Power = torque \times angular \quad velocity

\omega = \frac{2 \pi}{T} = 2 \pi f

The 1st Law of Thermodynamics

The first law of thermodynamics states:

Energy can only be converted: it can neither be created nor destroyed.

First law of thermodynamics, laws of thermodynamics, 1st law of thermodynamics

This means that in any process, 1-2, where there is a heat transfer to the system and a work transfer from the system, the difference in these transfers must equal the total change in energy of the system:

Q_{12} - W_{12} = U_2 - U_1

Remember the sign conventions for Q and W are opposite.

For a cycle, there can be no overall change in energy in the system at the start and end. Therefore:

\Sigma Q - \Sigma W = 0

Where heat and work transfers are summed respectively:

\Sigma = Q_{12} + Q_{23} + Q_{34} + ...

\Sigma W = W_{12} + W_{23} + W_{34} + ...

Adiabatic & Isothermal Processes

In a reversible process, assuming we can measure the changes in pressure and volume, we can calculate the work output as PΔV. Applying this to the first law gives:

\Delta Q - P \Delta V = \Delta U

We only know one quantity in the equation: ΔQ and ΔU are both unknown, and as such we either need to define one of them or use the second law.

These are two ways in which we can define one of the unknown properties above.


Adiabatic process, thermodynamics adiabatic process, adiabatic P-V diagram

In an adiabatic process, there is no heat change:

\Delta Q = 0

This then fully defines the process, and the first law can be used to find the change in internal energy:

\Delta W = -P \Delta V = \Delta U

From this, we get the following relationships for an ideal gas in an adiabatic process:

\frac{T_2}{T_1} (\frac{V_2}{V_1})^{\gamma - 1} = 1 \quad TV^{\gamma - 1} = constant

\frac{P_2}{P_1} (\frac{V_2}{V_1})^{\gamma - 1} = 1 \quad PV^{\gamma} = constant


Isothermal process, isothermal thermodynamic process, isothermal p-v diagram, isotherms, constant temperature process

Instead of insulating the system, a thermal reservoir is attached. This means the fluid is always at the same temperature (that of the reservoir), and as such there is no change in internal energy:

\Delta U = 0

Again, the process is now fully defined, and so the changes in heat and work are given by the first law:

\Delta Q = \Delta W

Note that adiabatic processes are far quicker than isothermal processes (see the gradients of the curves)

This is because the equation of the curve is given as:

P = \frac{mRT}{V}

  • In an adiabatic process, T decreases as V increases
  • In an isothermal process, T is constant so only V increases
  • The total energy of a system is given as the sum of the internal, kinetic, and potential energies
  • For a closed system, the change in total energy = change in internal energy
  • Heat transfer is the transfer of energy between a system and its surroundings due to a temperature difference
  • A heat transfer to the system (heating up) is positive, a heat transfer from the system (cooling down) is negative
  • Work transfer is the transfer of energy between a system and its surroundings that is not due to a temperature difference
  • According to the first law, energy must be conserved: ΔQ – ΔW = ΔU
  • For a cycle, ΣU – ΣW = 0