Thermodynamics

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First law of thermodynamics
work done by the system
The change in the internal energy of a system
depends only on the net heat transferred to the
system and the net work done by the system, and
is independent of the particular processes
involved.
The equation is deceptively simple One of the
forms of the general law of conservation of
energy. BUT! Be careful about the sign
conventions. Positive Q is heat transferred to
the system. Positive W is work done by the system.
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Work done and heat transferred our major
concerns!
Work done by the gas
P pressure of the gas Dx displacement of
the piston, A area of the piston
The work is positive when the gas expands!
Differential form
Integral form also good for varying pressure
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Work done and heat transferred our major
concerns!
Work done by the gas
P (varying) pressure of the gas dV
differential volume change
P
V
Area under curve on a P-V diagram
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Multiple ways to get (quasi-statically !) from an
initial to a final state
Is work going to be the same for different
processes?
NO!
NO!
Is heat going to be the same?
Is internal energy going to be the same?
YES!
Internal energy is a function of state and will
be the same as well as it variation between the
states.
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Isothermal process T const
Isotherm
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Isothermal process - T const Example 1
A scuba diver at a depth of 25m, P 3.5atm,
exhales bubbles 8.0mm in radius. How much work
is done by each bubble as it expands while it
rises to the surface?
Solution From the ideal gas law we know for an
isothermal process that the pressure and volume
are inversely related.
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Isothermal process - T const Example 2
An ideal gas expands to 10 times its original
volume, maintaining a temperature of 440K. The
gas does 3.3kJ of work.
(a) How much heat does the gas absorb?
(b) How many moles of gas are there?
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Constant volume (isochoric) process V const.
n number of moles of the gas Cv molar
specific heat at constant volume heat capacity
of one mole of the gas in an constant volume
process.
(Compare with )
Why bother introducing a new parameter? For a
gas, per mole is more convenient.
Measuring Cv we learn about internal energy of
the gas as a function of temperature!
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Constant volume - (isochoric) process
From the kinetic theory of gasses, we know that
the kinetic energy/molecule is
If kinetic energy is the only form of internal
energy then
How much heat is required to raise the
temperature of 500g of He by 10oC?
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Isobaric processes P const.
P
V
Since the pressure of the gas remains constant,
calculation of the work it does is particularly
simple.
What about internal energy and heat? Is T2lt or
gtT1?
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Isobaric processes P const.
What about internal energy and heat?
P
V
From the 1st law and equation for Cv
Ideal gas at constant pressure
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Isobaric processes P const.
P
V
Molar specific heat at constant pressure
(definition)
Why is specific heat at constant pressure higher
than at constant volume?
What is the specific heat at constant pressure
for He gas?
Why was there no difference in specific heats for
solids?
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Review of our Thermodynamic Processes
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Adiabatic processes no heat transfer Implies
thermal insulation or process happens quickly.
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Adiabatic processes no heat transfer.
Positive work, W, is done by the expanding gas at
expense of reduction of its internal energy
Since there is no heat supplied from the
outside to replenish the gas energy the
temperature declines.
Work can be expressed in terms of DT or D(PV)
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Adiabatic processes no heat transfer.
We now know the work done by an ideal gas during
an adiabatic expansion, but how do pressure and
temperature behave?
From the ideal gas law
From the 1st law
As an exercise for the student/professor
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Adiabatic processes no heat transfer.
A gas with g1.4 at 105Pa occupies 5L. It is
compressed adiabatically to 2.5L.
(a) What is its final pressure?
(b) What is the ratio of the final and initial
temperature?
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Adiabatic processes no heat transfer.
A gas with g1.4 at 105Pa occupies 5L. It is
compressed adiabatically to 2.5L.
(a) What work is required to compress the gas?
Why is the work negative?
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Adiabatic processes
Blue arrow the gas is expending, does a
positive work and cooled down (goes to lower
isotherms)
Red arrow the gas is contracting, driven by
some external forces. It does negative work and
heats up (goes to higher isotherms.)
An isothermal process is described by
An adiabatic process is described by
g Cp/Cv a constant, where 1lt g lt 2 It may
be different for different gases!
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How does it happen that two equations may work
for the same idea gas?
Are they compatible with the ideal gas law?
Yes, they are both compatible! Because the ideal
gas law has 3 variables, and P as a function of V
in a particular process depends on what happens
to the temperature!
Isothermal
Adiabatic
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Examples of isothermal and adiabatic expansion
Consider expanding .06 mol of gas from 2atm to
1atm. Starting with T300K. Find the final
temperature and volume for both processes.
Assume
The initial volume is
For isothermal expansion the volume doubles. For
adiabatic expansion
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Fire Syringe A small wad of cotton bursts into
flame when the air in a narrow tube is rapidly
compressed.
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4 processes between two isotherms. How do you
order them in terms of heat transferred to the
gas?
Things to remember U depends on T only!
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What is the heat transferred to the gas in
process 3?
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Specific Heats of an Ideal Gas
From kinetic theory we showed the average kinetic
energy per molecule is
For n moles the internal energy due to KE is
This means we can find Cv
For inert gases we have
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Specific Heats of an Ideal Gas
However for diatomic gases, nitrogen, oxygen,
etc
Whats up??
Internal energy may be more than just
translational kinetic energy!
Equipartition Theorem
In thermodynamic equilibrium, the average energy
per molecule is ½ kT for each degree of freedom
This means Cv½RT for each degree of freedom For
a diatomic molecule there are 5 degrees of
freedom
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Summary of the First Law of Thermodynamics
First Law of Thermodynamics
Thermodynamic processes, Quasi-static - work is
given by
Isothermal constant temperature Isochoric
constant volume Isobaric constant
pressure Adiabatic no heat transfer
Equipartition theorem 1/2kT average molecular
energy for each degree of freedom