Download Chapter 19 – The First Law of Thermodynamics

Chapter 19 – The First Law of Thermodynamics
I.
Introduction – We have been looking at how various systems behave as a result of heat being
exchanged. Now we will first look at how work affects a thermodynamic system, then examine the
energy relationships in a thermodynamic system.
II.
Work
A.
Mechanical work – look at the work done by a system (gas) as its volume changes.
Consider a cylindrical container with a moveable, frictionless piston with a cross-sectional area A.
Within the container is a gas at a pressure (absolute) p. Suppose that the piston is pushed upward a
distance dx.
dx
p
 
The work done by the gas is: W  F  ds  Fx dx 

W
Vf


 pAdx .
Adx is the volume dV, so
pdV or dW  pdV
Vi
When a graph of pressure p versus volume V is drawn, the graph is referred to as a pV diagram.
Pressure, p
Vi
Vf
Volume, V
If the volume increases, then work is done by the system and the work is positive.
If the volume decreases, then work is done on the system and the work is negative.
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III. Types of thermodynamic processes and work done in the processes
piston
Again consider an ideal gas inside a
cylinder that is fitted with a piston.
gas
A.
Isobaric process
p
V
B.
Isochoric or isovolumetric process
p
V
C.
Isothermal Process
p
V
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IV. Combine the Effects of Heat Transfer and Work
Consider a container that is fitted with a piston filled with a gas at some initial pressure, volume and
temperature. An amount of heat Q will be added to the system (gas) and the energy changes will be
examined.
Before the heat is added, how much energy and what type of energy is associated with the system?
A.
Add heat to the system while keeping the volume constant. Where does the energy go? Note that
the temperature of the gas must change.
B.
Add heat to the system and keep the temperature constant. Where does the energy go? (Remember
that the temperature is directly proportional to the translational kinetic energy, TKE.)
C.
Add heat to the system, in general. First Law of Thermodynamics.
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D.
Interesting Note: If a system changes from an initial state i to a final state f along different paths (e.g.,
Path A and Path B), the change in internal energy will be the same along those paths. And, in fact, all
paths that go from i to f. That is, U = Uf - Ui .
From the first law, that means that Q – W is also constant going from i to f.
p
Path A
i
f
Tf
Path B
Ti
V
V.
The First Law of Thermodynamics as applied to various thermodynamic processes:
U = Q - W and
A.
dU = dQ - dW
Isochoric process, V = constant and dV = 0
0

U  nCV T  pdV
since Q  nCV T
U  nCV T
0
dU = dQ - pdV
since dQ = nCVdT
dU = nCVdT
This result represents the increase in the internal energy of molecules as a result of the temperature
change T or dT. This result is valid for all processes involving an ideal gas.
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B.
Isobaric process, p = constant and dp = 0
C.
Isothermal process, T = constant and dT = 0
D.
Adiabatic process (no heat exchanged), Q = 0 and dQ = 0
E.
Cyclic process (start and end in the same state)
F.
Some examples:
1.
p
a
iaf: Q = + 50 J
f
W = + 20 J
ibf: Q = + 36 J
i
b
V
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a.
Find the work done going from i to b to f, Wibf .
b. If the work done going from f to i, Wfi = - 13 J , then find the heat exchanged going from f
to i, Qfi .
c.
d.
2.
If the internal energy at point i is Ui = 10 J , then find the internal energy at f, Uf .
If the internal energy at point b is Ub = 22 J , then find the heat exchanged going from i to b,
Qib , and from b to f, Qbf .
For the following data, find the heat exchanged
going from state C to A, QCA : QAB = 20 J, QBC = 0,
and WBCA = + 15 J.
p
B
A
C
V
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3.
p (atm)
Find the net heat transferred in the cyclic
process.
30
20
10
0
1
2
3
4
V (L)
VI. Heat Capacities for Ideal Gases
A.
Monatomic Gas
1.
Remember that in an isochoric process, the molar heat capacity for a monatomic gas is
CV 
2.
3
R
2
What is the molar heat capacity of the gas when heat is added isobarically? Add heat while
keeping the pressure constant. In this case the added heat energy goes into increasing the
kinetic energy (internal energy) of the molecules and in doing work on the environment
(volume increases).
Heat Added = Heat (KE) + Heat (Work)
dQ = dU + dW = nCVdT + pdV
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3.
Ratio of heat capacities, 
B.
Remember the value of the molar heat capacity at a constant volume, CV, for molecules with f
degrees of freedom.
C.
Experimental values of  at 300 K
Monatomic
Helium
Argon
1.67
1.67
Diatomic
Hydrogen
Oxygen
Nitrogen
Chlorine
1.41
1.40
1.39
1.35
Polyatomic
CO2
SO2
C2H6
NH3
(C2H5)2O
1.30
1.29
1.20
1.31
1.07
VII. Examine the adiabatic process again, and find relationships between the state variables.
A.
Start with the first law and the equation of state of an ideal gas
dU = dQ - dW and
pV = nRT
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Example: One mole of a monatomic gas expands adiabatically from 10 atm at 0 oC to a pressure of 2
atm. Find the initial volume of the gas, the final volume, and the final temperature.
B.
Work done in an adiabatic process
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VIII. Examples of cyclic processes. Complete the following tables: p
Example 1. Shown in the diagram is a cyclic process for n
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moles of a diatomic gas with heat capacities CV =
R and
2
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Cp =
R.
2
1
2
dT = 0
State
1
2
3
p(Pa)
V(m3)
3
T (K)
100
4
1
Process
V
4
Name
Q (J)
W (J)
U (J)
S (J/K)
1 to 2
2 to 3
3 to 1
1 to 2 to 3 to 1
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Example 2. A cyclic process is shown for a diatomic gas with a
ratio of heat capacities  = 1.40.
State
p (atm)
V (L)
T (K)
A
B
C
2
1
2
273
p
A
B
C
V
Process
Name
Q (cal)
W (cal)
U (cal)
S (cal/K)
A to B
B to C
C to A
A to B to C to A
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