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Then we must fix the convention on the symbol
for heat and work:
We will always denote
by Q the quantity of heat added to the system
by W the mechanical work done by the system
Therefore Q and W are understood as algebraic
values, they can be positive, negative or zero.
system
surroundings
system
surroundings
system
surroundings
system
surroundings
Q > 0
Q < 0
W > 0
W < 0
Work is done by the system
Work is done on the system
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1.2 Calculation of work done during volume changes:
A typical example of a thermodynamic system
is an amount of gas enclosed in a cylinder
with a movable piston. (Such a system is the
central part of heat engines: locomotive,
engine of a car, refrigerator,…).
dx When a gas expands, it does work on its
environment. For a small displacement dx,
the work done by the gas is:
dW
by
= F dx = p A dx = p (A dx)= p dV
A
Consider the expansion of gas of from an initial state (with the volume
V
1
) to a final state (the volume V
2
). The system (gas) passes through
a series of intermediate states. We assume the changes of states are
slow enough, then every intermediate state can establish equilibrium,
and has determined values of p, V, T.
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Note that when the gas expands, V
2
> V
1
→ W
by
> 0 , and when
the gas is compressed, V
2
< V
1
→ W
by
< 0 (it means that the
surroundings does work on the gas).
In a p-V diagram, the equilibrium intermediate states are represented
by the points on a curve, and the work is represented as the area under
the curve
The work done by the gas during the whole change V
1
→ V
2
is
2
1
V
V
by
pdVW
p
V
V
2
V
1
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1.3 Paths between thermodynamic states:
When a thermodynamic system changes from an initial state to a final
state, it passes through a series of (equilibrium) intermediate states.
However, with the same initial and final states, the system can pass in
very different ways. On a P-V diagram, every way corresponds to a
curve which is called the path between thermodynamic states.
Examples: Two different paths between the states 1 and 2 :
V
p
p
V
1
2
3
1
2
4
1 → 3 : keep the pressure constant
at p
1
while the gas expands
to the volume V
2
3 → 2 : reduce the pressure to p
2
at
constant volume V
2
p
1
V
2
p
1
p
2
p
2
1 → 4: reduce the pressure
at the constant volume V
1
4 → 2: keep the pressure
constant at p
2
while the gas
expands to the volume V
2
V
1
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It is important to remark that with the same intial and final states:
The work done by the system depends on the intermediate states,
that is, on the path,
Like work, the heat which the system exchanges with the
surroundings depends also on the path.
p
V
1
2
p
V
1
2
Examples:
In an isothermal expansion of the gas
we must supply an input heat to keep
constant temperature
Gas can expand in an
container which is isolated
from surroundings (no heat
input)
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§2. The first law of thermodynamics:
2.1 Internal energy of a system:
The internal energy of a system is the energy that the system owns.
We can define:
(Note that the internal energy does not include potential energy arising
from the interaction between the system and its surroundings, for
example, system and gravitaitonal field).
Internal energy = ∑kinetic energies of constituent particles
+ ∑potential energies between them
For an ideal gas we know how can calculate the internal energy. But
for any real system, the calculation of the internal energies by this way
would be very complicated.
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We have another way. Practically, in the study of thermodynamical
processes, we can determine not just the interal energy U , but the
change in internal energy ΔU .
We can choose by convention the internal energy of the system
at any reference state, and then knowing ΔU we can determine
U at all other states.
(Recall that the potential energy of a particle in a gravitational field,
or the potential energy of a charge in the static electric field are
defined with the precision to an adding constant).
Having the concept of the internal energy, we can formulate
the first law of thermodynamics
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2.2 Formulation of the first law of thermodynamics:
Consider a change of state of the system from an initial value U
1
to a final value U
2
, then ΔU = U
2
– U
1
.
If the change is due to the addition of a quantity of heat Q with
no work done → the inernal energy increases, and ΔU = Q .
If the system does work W by expanding and no heat is added,
the internal energy decreases, we have ΔU = - W
The first law of thermodynamics states that when both heat transfer
and work occur, the total change in internal energy is
ΔU = Q - W
Note: Always remember the convention on the signs of Q and W
given before !!!
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§3. Kinds of thermodynamic processes:
We know that there are many different paths between thermodynamic
states. We will study four specific kinds of thermodynamic processes
which are important in practical applications.
3.1 Adiabatic process:
Definition: Adiabatic process is defined as
one with no heat transfer
into or out of a system, Q = 0.
Examples:
Gas in a container which is surrounded by a thermally isolating
material
A expansion (or compression) of gas which takes place so quickly
that there is not enough time for heat transfer.
From the 1
st
law: ΔU = U
2
– U
1
= - W (adiabatic process)
p
V
1
2
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3.2 Isochoric process:
Definition: This is a constant-volume process.
Example: A gas in a closed constant-volume
container.
When the volume of a thermodynamic system is
constant, it does no work on its surroundings
W = 0
From the 1
st
law: ΔU = U
2
– U
1
= Q (isochoric process)
V
p
1
2
Since the system does no work → all the energy (heat) added
remains in the system → the iternal energy increases.
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