First Law of Thermodynamics

A thermodynamic system is a specific, macroscopic region of space or quantity of matter chosen for thermodynamic study, separated from its surroundings by a real or imaginary boundary.

Open system: exchanges both mass and energy (heat or work) with its surroundings. Example: boiling water in an open pan, where steam escapes and heat is transferred.

Closed system: exchanges energy but not mass. Example: gas sealed in a cylinder with a movable piston; energy is transferred as heat or by piston movement but the gas does not escape.

Isolated system: exchanges neither mass nor energy. Example: hot coffee in a perfectly insulated, sealed thermos flask.

Work done by the system (expansion): the system pushes on the surroundings, for example a gas pushing a piston outward. \(dV\) is positive, so the work is taken as positive (\(+W\)) and the system loses internal energy to perform it.

Work done on the system (compression): the surroundings push on the system, for example a piston pushed inward to compress a gas. \(dV\) is negative, so the work is taken as negative (\(-W\)) and the internal energy of the system increases.

For a constant-pressure process:

An indicator diagram (P-V diagram) is a graph of the pressure \(P\) of a thermodynamic system on the y-axis against its volume \(V\) on the x-axis.

The work done by an expanding gas is

which is equal to the area under the P-V curve between the initial volume \(V_1\) and the final volume \(V_2\). When the volume increases (left to right) the work is positive.

Internal energy: internal energy \(U\) is the sum of the microscopic kinetic energies (translational, rotational, vibrational) and potential energies (due to intermolecular forces) of all the molecules of a system. It is a state function.

Relation with heat and work: the first law, \(dQ=dU+dW\), connects them. Heat added to a system raises its internal energy and does external work.

Meaning of \(dU=-dW\): it follows when \(dQ=0\), that is, in an adiabatic process. If the gas expands and does work (\(dW>0\)), the work is done at the expense of the internal energy, so \(dU\) is negative and the gas cools.

Ideal and real gas: an ideal gas has no intermolecular forces, so its internal energy is purely kinetic and depends only on temperature. A real gas has intermolecular forces, so its internal energy has both kinetic and potential parts and depends on temperature and volume.

Statement: if an amount of heat \(dQ\) is supplied to a system, it is used in two ways: to increase the internal energy of the system (\(dU\)) and to do external work (\(dW\)) by the system.

Significance: it establishes the equivalence of heat and mechanical work, shows that heat is a form of energy, and states that every thermodynamic process obeys conservation of energy.

Yes. The first law is the principle of conservation of energy applied to thermodynamic systems. Energy can neither be created nor destroyed; it only changes from heat into internal energy and mechanical work.

Direction of heat flow: it does not tell in which direction heat flows, for example why heat flows from hot to cold and not the other way.

Spontaneity: it gives no information on whether a given process is possible or spontaneous.

Extent of conversion: it does not say how much heat can be converted into work; the second law is needed to show that complete conversion in a cycle is impossible.

Solids and liquids change very little in volume on heating, so one specific heat is enough. A gas, however, can change greatly in volume or pressure. Heated at constant volume, all the heat goes into internal energy. Heated at constant pressure, the gas also expands and does external work. Different amounts of heat are needed for the same temperature rise in the two cases, so a gas has two principal specific heats: \(C_v\) (constant volume) and \(C_p\) (constant pressure).

Consider 1 mole of an ideal gas.

At constant volume: \(dQ_v=C_v\,dT\). Since \(dW=P\,dV=0\), the first law gives \(dU=C_v\,dT\).

At constant pressure: \(dQ_p=C_p\,dT\), and the gas does work \(dW=P\,dV\). The first law gives

The internal energy of an ideal gas depends only on temperature, so \(dU=C_v\,dT\) is the same in both cases:

From \(PV=RT\) at constant pressure, \(P\,dV=R\,dT\). Therefore

At constant volume all the heat supplied raises the internal energy (temperature). At constant pressure the gas also expands and does external work, so extra heat must be supplied for this work in addition to the heat that raises the internal energy. Hence \(C_p>C_v\).

Isothermal: temperature remains constant (\(\Delta T=0\)). It occurs slowly in a highly conducting vessel, and \(PV=\text{constant}\).

Adiabatic: no heat is exchanged with the surroundings (\(\Delta Q=0\)). It occurs rapidly or in a well-insulated vessel, and \(PV^{\gamma}=\text{constant}\).

Isobaric: pressure remains constant (\(\Delta P=0\)), for example heating a gas in a cylinder with a freely moving, weight-loaded piston.

Isochoric: volume remains constant (\(\Delta V=0\)), for example heating a gas in a rigid sealed container.

Starting from the same initial state \((P_1,V_1)\):

Figure: P-V diagrams of the four thermodynamic processes from the same initial state

Isobaric is a horizontal straight line (slope \(=0\)). Isochoric is a vertical straight line (slope \(=\infty\)). Isothermal is a rectangular hyperbola sloping downward with slope \(\dfrac{dP}{dV}=-\dfrac PV\). Adiabatic is also a downward curve but with slope \(\dfrac{dP}{dV}=-\gamma\dfrac PV\); since \(\gamma>1\), it is steeper than the isothermal curve.

An adiabatic process is a thermodynamic change in which no heat enters or leaves the system (\(dQ=0\)).

For \(n\) moles, the first law with \(dQ=0\) gives

Differentiating \(PV=nRT\):

Substituting in (1) and using \(R=C_p-C_v\):

Since \(\dfrac{C_v}{C_p-C_v}=\dfrac1{\gamma-1}\), multiplying through by \((\gamma-1)\) gives

Dividing by \(PV\) and integrating:

Adiabatic, so \(Q=0\), and the work done by the gas is \(W=+1200\ \mathrm{J}\):

Final temperature:

Work done:

Isothermal work. \(W=\displaystyle\int_{V_1}^{V_2}P\,dV\) with \(P=\dfrac{nRT}{V}\) and constant \(T\):

Adiabatic work. For an adiabatic process \(PV^\gamma=K\), so \(P=KV^{-\gamma}\):

Using \(K=P_1V_1^\gamma=P_2V_2^\gamma\):

The heat capacity ratio is the ratio of the molar heat capacity of a gas at constant pressure to that at constant volume:

For a gas with \(f\) degrees of freedom, \(\gamma=1+\dfrac2f\).

As atomicity, and with it the degrees of freedom, increases, \(\gamma\) decreases.

Reversible process: an idealised process that occurs infinitely slowly so that the system is in thermal and mechanical equilibrium with its surroundings at every stage. It can be retraced along the same path, leaving no net change in the system or the universe.

Irreversible process: a real process occurring at a finite rate, usually involving friction or rapid heat transfer. The system is not in equilibrium at each step and cannot be restored to its original state without a permanent change in the surroundings (an increase in entropy). All natural processes are irreversible.