Thermodynamics

Heat

THERMODYNAMICS (return to CONTENTS)

INTERNAL ENERGY, HEAT AND WORK (return to start of page)

Thermodynamics deals with processes which involve energy changes as a result of heat flow to or from a system and/or work done on or by a system.

Often the ‘system’ considered is a fixed mass of gas separated from its surroundings by a cylinder and a piston:

The energy in a system is called its internal energy.

The state of a gas at a given moment is defined in terms of the particular values of mass, volume, pressure, temperature and internal energy it has at that moment.

Heat and work

Heat and work are terms used to describe energy in the process of transfer:

Heat is energy that flows from one body to another because of a temperature difference between them
Work is energy that is transferred by a force moving its point of application over a distance

The internal energy of a system can be changed by heating and/or working.

The term ‘internal energy’ of a system is preferred to the term ‘heat content’, since, for example, in the above system, if the piston is pushed in a little, work is done on the gas, and its temperature and internal energy will rise, but no heat has entered the gas.

The zeroth law of thermodynamics (this law was formally stated after the first one)

If a hot and cold body are brought into contact, after a while all heat transfer between them will stop. The bodies are then said to be in thermal equilibrium - they share the common property which we call temperature.

The zeroth law says that if bodies A and B are each separately in thermal equilibrium with body C, then A and B are in thermal equilibrium with each other

Thus, if C is a thermometer and reads the same when in contact with A or B, then A and B are at the same temperature and are in thermal equilibrium - if A and B were put in contact with each other, no heat would flow between them

The first law of thermodynamics

The internal energy of a system can be:

increased if heat flows into it or work is done on it
decreased if heat flows out of it or work is done by it

This is the first law of thermodynamics (it is a consequence of the principle of conservation of energy).

The signs assigned to the various quantities is a matter of convention. Here we will specify:

Work done by an expanding gas

Gas pressure, P = force/area, so force = pressure * area, i.e. F = PA.

The piston is held in position by the force PA exerted by the gas and the external force F.

Suppose that the volume changes by a finite amount. A graph such as the following of P against V during such a change is called an indicator diagram:

The area between the graph and V₁ to V₂on the axis can be divided into strips like the one shown. The sum of the areas of all such strips equals the total work done. Hence, the total work done in the finite expansion from V₁ to V₂equals the area below the graph.

If pressure P is constant, the change is called an isobaric change, in which case the indicator diagram looks like:

Thus, for an isobaric change, the work done is given by:

The equation applies equally if the gas is compressed at constant pressure, in which case W is the work done on the gas.

Reversible changes

To calculate work done using W = P(V₂- V₁) we assume that P is the same at every stage of the change. This implies that the system is in equilibrium (i.e. all its parts are at the same temperature and pressure) at every instant of the change. Thus, during a change we regard a system as passing through an infinite series of states of equilibrium. Since it would be possible to take the system back through the same set of equilibrium states, such a change is said to be reversible.

PRINCIPAL HEAT CAPACITIES OF A GAS (return to start of page)

Recall that the heat capacity of something is the amount of heat required to raise its temperature by 1⁰C (or 1K).

Consider heat being supplied to a gas trapped in a cylinder by a piston, as in the earlier diagram. The heat required to warm the gas by 1⁰C will depend upon how much the gas expands, since heat is needed to:

increase the internal energy of the gas, and
provide the energy for any external work done by the gas in expanding and pushing back the outside atmosphere

Now, between the original position of the piston and any other position there are an infinite number of possible intermediate positions, and so there are an infinite number of different amounts of external work that might be done, and so a gas has an infinite number of possible heat capacities. However, only the simplest two are important, called its principle heat capacities. These relate to:

constant volume, in which the piston is not allowed to move at all
constant pressure, in which the piston is allowed to move freely, keeping the gas pressure equal to the outside pressure

The term ‘heat capacity’ refers to a particular object or system. To specify the amount of a substance, we refer to one mole or unit mass of that substance, and then we use the terms ‘molar heat capacity’ or ‘specific heat capacity’.

The principle molar heat capacities of a gas are defined as:

The molar heat capacity at constant volume is the heat required to produce a unit temperature rise in one mole of a gas when the volume is kept constant (denoted by C_v)
The molar heat capacity at constant pressure is the heat required to produce a unit temperature rise in one mole of a gas when the pressure is kept constant (denoted by C_p)

In the similar definitions for the principle specific heat capacities, c_v and c_p, ‘unit mass’ replaces ‘one mole’.

Derivation - relationship between C_p and C_v.

We consider a fixed mass of gas trapped in a cylinder by a frictionless piston. We specify ‘frictionless’ because if the piston moves, no energy is ‘lost’ due to friction.

The gas is heated till its temperature rises by 1⁰C (= 1K), with:

Recalling the equation for the first law of thermodynamics:

Since R is a positive constant, C_P > C_V.

ISOTHERMAL AND ADIABATIC CHANGES (return to start of page)

Isothermal change

This is a change that occurs at a constant temperature

Isothermal changes are typically slow changes, since heat has to have time to enter or leave the system to keep its temperature constant.

So, in an isothermal expansion, the heat supplied to the gas equals the work done by the gas (and conversely for an isothermal compression).

Also, since the temperature is constant, the expansion curve follows a Boyle’s law P-V curve:

The area enclosed by the graph and the axes, between volumes V₁and V₂, equals the work done during the volume change.

Adiabatic change

This is a change that occurs without heat entering or leaving the gas

Adiabatic changes are typically rapid changes, in which heat does not have time to enter or leave, for example:

air escaping an exploding tyre
sounds waves - rapid compressions and expansions of air as the sound wave travels

So, in an adiabatic expansion, all the work done is at the expense of the internal energy of the gas, and so the temperature of the gas falls.

Conversely, in an adiabatic compression, work is done on the gas by an external agent, increasing the internal energy and so the temperature of the gas rises.

Notice that the slope (gradient) of the adiabatic curve through any point is greater than the slope of the isothermal curve through the same point.

Boyle’s law (PV = a constant) assumes that the temperature is constant, and so does not apply to an adiabatic change.

HEAT ENGINES (return to start of page)

A heat engine converts heat into mechanical work.

Heat engines operate by taking some ‘working substance’ (e.g. the gas in the cylinder of an internal combustion engine) around a repeating cycle, in which:

heat is taken in at a high temperature
work is done (when the gas expands, pushing back a piston)
some heat is rejected at a lower temperature

The basic process can be represented by:

Efficiency of a heat engine

It can be shown that for an ideal heat engine:

The efficiency given by this expression is the maximum theoretically achievable for the temperatures of the source and sink. Real heat engines are much less efficient.

For example, suppose a steam turbine is driven by steam at 600⁰C which is exhausted at 100⁰C. Adding 273 to each temperature to get kelvins, then:

The actual efficiency is about 30%. The efficiency of a petrol engine is only about 20%.

Carnot’s ideal heat engine (1824)

Carnot considered an ideal heat engine, in which a working substance is taken reversibly through a cycle represented by:

Along AB an isothermal expansion occurs at temperature T_H, heat Q₂ being absorbed
Along BC an adiabatic expansion occurs, and the temperature falls to T_C
Along CD an isothermal compression occurs at temperature at T_C, heat Q₁ being rejected
Along DA an adiabatic compression occurs, and the temperature rises back to T_H

At the end of the cycle the substance is in the same state it was initially, i.e. the same P, V and T and internal energy. However, during the cycle, the net external work done equals Q₂ - Q₁, and this equals the area enclosed by the 4 curves.

The petrol engine cycle (or Otto cycle)

This is a four-stroke cycle: down - up - down - up

The diesel engine cycle

In both the above cases, the fuel is burnt inside the cylinder, and so both are internal combustion engines.

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