Download Chapter 1: The first law of thermodynamics

Chapter 1: The first law of thermodynamics
1.1 Fundamental definitions
• Zeroth law of thermodynamics: If systems A and B are separately in equilibrium with system
C, then systems A and B are in equilibrium with each other.
• The amount of energy transfer needed to cause the temperature to rise by one temperature unit
is called the heat capacity.
1.3 Different aspects of equilibrium
There are three different aspects of equilibrium between systems.
1) The first is thermal equilibrium, where there is a flow of energy between systems which
eventually balances and the flow stops.
2) The second is called mechanical equilibrium, where the volume of each system changes until
a balance is reached and then the volume changes stop.
3) The third is chemical equilibrium, where there is a flow of particles between the systems until
a new chemical balance is reached and the flow stops.
We say that the system is in thermodynamic equilibrium when thermal, mechanical, and chemical
equilibrium have been reached. The system then has a well-defined pressure, temperature, and
chemical potential.
1.4 Functions of state
Whenever a quantity only depends on the present values of macroscopic variables such as the pressure
and volume we say that the quantity is a function of state. Therefore, for an ideal gas in equilibrium,
the system’s temperature is a function of state ( θ = F ( P,V ) ).
A quantity, dG, is said to be an exact differential if it only depends on the difference in the function of
state between two closely spaced points and not on the path between them.
1.5 Internal energy
When the state of an isolated system is changed by the performance of work, the amount of work
needed is independent of the means by which the work is performed.
By observing that a system can be taken through any attainable intermediate state, by any series of
process we choose, with the final equilibrium state being the same, we introduce a quantity called the
internal energy of the system. We choose the initial energy, denoted by U, so as to ensure that the total
energy of the system is constant.
The total work done is independent of the path chosen from the initial to the final equilibrium states; it
follows that the quantity U is a function of the initial and final states only. The quantity U is therefore
a function of state.
Assume now that the system is not thermally isolated. To preserve the law of conservation of energy
we choose to write the change in internal energy from its initial value Ui to its final value Uf as
ΔU = U f − U i = W + Q
where Q is defined as the heat which is added to the system, and W is the work done on the system.
The extra term, Q, is included so that we can preserve the law of conservation of energy for systems
which are not thermally isolated.
The first law of thermodynamics then says that energy is conserved if heat is taken into account.
1.6 Reversible changes
A reversible process is defined as one which may be exactly reversed to bring the system back to its
initial state with no other change in the surroundings.
A quasistatic change is one which is made in such a way that the system remains in equilibrium at all
times. A reversible process involving a large change proceeds through a series of quasistatic states.
For reversible processes involving a small change in volume, the first law of thermodynamics can be
written as
dU = dQrev + dW = dQrev − PdV
The total work done is the integral of − PdV between the initial and final states; thus
V2
ΔW = − ∫ PdV
V1
Work is not a function of state: it depends not only on the initial and final equilibrium states but also
on the path taken. Furthermore heat is not a function of state.
1.7 Enthalpy
The quantity U + PV is called the enthalpy; it is denoted by the symbol H. Since U, P, and V are
functions of state, it is clear that enthalpy is a function of state as well.
1.8 Heat capacities
When a tiny amount of heat energy is added to a system it causes the temperature to rise. The heat
capacity is the ratio of the heat added to the temperature rise. Thus
C=
dQrev
dθ
However, this ratio is not well defined because it depends on the conditions under which the heat is
added (constant pressure, constant volume etc.). For one mole of an ideal gas
CP = CV + R
1.9 Reversible adiabatic changes in an ideal gas
An adiabatic change is defined as a change in which no heat enters the system. For reversible
adiabatic changes we have
PV γ = constant,
where γ =
CP
.
CV