ESCI 341 – Atmospheric Thermodynamics

Lesson 5 – Thermodynamic Processes

 

References:   An Introduction to Atmospheric Thermodynamics, Tsonis

Introduction to Theoretical Meteorology, Hess

Physical Chemistry (4th edition), Levine

Thermodynamics and an Introduction to Thermostatistics, Callen

 

Reading:  Tsonis, Section 4.6

Hess, Section 3.5 (e-reserve)

 

ISOTHERMAL PROCESSES

l      If a process is isothermal (dT = 0) then for an ideal gas the first law becomes

l      For an ideal gas we can substitute for p from the ideal gas law to get

which integrates to

.

ISOCHORIC PROCESSES 

l      If a process is isochoric (constant volume) then the first law for an ideal gas becomes

.

l      This can be integrated to get (assuming cv is constant)

.

 

ISOBARIC PROCESSES

l      For an isobaric process, dp = 0.  Therefore the first law for an ideal gas becomes

which integrates to

.

 

ADIABATIC PROCESSES

l      An adiabatic process is one in which there is no heat transfer (dq = 0).

l      The first law of thermodynamics for an adiabatic process is

o        For an ideal gas, the first law for an adiabatic process is

l      If we start with the first form of the first law for an ideal gas (the one involving cv) and substitute for pressure from the ideal gas law, we get

.

o        Integrating this gives

which can also be written as

o        We’ve previously shown that .  Therefore, we can write

and defining the ratio  we get

o        Using the ideal gas law, this equation can also be written as

or

l      These three relations are known as Poisson’s relations (note that the constant is not necessarily the same in each equation.

  Poisson’s relations

l      Poisson’s relations relate T, p, and a in ideal gases undergoing quasi-static adiabatic processes.  If you know the initial values of two of these variables, and one of their final values, you can find the other two final values by using these relations.

l      It is important to realize that Poisson’s relations are only valid for ideal gases undergoing quasi-static adiabatic processes!

 

POTENTIAL TEMPERATURE

l      Meteorologist’s most often make use of the Poisson relation between T and p.

l      Potential temperature (denoted as q ) is defined as the temperature an air parcel would have if it were moved dry-adiabatically to a reference pressure, p0, of 1000 mb.

l      From Poisson’s relation for T and p we get

.

l      If an air parcel undergoes an adiabatic process its potential temperature is conserved.

 

WORK IN AN ADIABATIC PROCESS

l      For an adiabatic process the change in internal energy is solely due to work done on or by the system, du = dw.

l      For an ideal gas, du = cvdT = dw.

o        Note that this is true regardless of whether the adiabatic process is quasi-static or not.

 

EXERCISES

1.  Show that for an isothermal process for an ideal gas

.

 

2.  For an isothermal process for an ideal gas, show that the work done by the system is

or

.

 

3.  a.  For an isobaric process show that

.

 

b.  Is this true for all gasses, or only ideal gasses?

 

 

 

4.  Starting with cpdT = dq +adp, derive the Poisson relation   (Show all steps!)

 

 

5.  A 1.5‑kg parcel of dry air is at a temperature of 15°C and a pressure of 1013 mb.

 

a.  How many moles of air are in the parcel? (The molecular weight of air is 28.96 g/mol)

 

b.  What is the volume of the parcel?

 

c.  What is the specific volume of the parcel?

 

d.  If 50 KJ of heat are added to the parcel while its volume is held constant, what is the new temperature of the parcel? (The specific heat of air at constant volume is 717 J-kg-1-K-1).

 

6.  An parcel of dry air is at a temperature of 15°C and a pressure of 1013 mb.  Heat is added to the parcel to cause it to expand.  It expands at constant pressure to 1.5 times its original volume. 

 

a.  What is the new temperature of the parcel?

 

b.  How much work (per unit mass) was done by the parcel during this expansion?

 

c.  What was the change in specific internal energy of the air parcel?

 

d.  What was the amount of heat per unit mass that was added to the air parcel?

 

 

7.  An air parcel is at a temperature of 15°C and a pressure of 1013 mb. Heat is added to the parcel to cause it to expand.  It expands at constant temperature until its volume is 1.5 time it original volume.

 

a.   What is the new pressure of the air parcel?

 

 

b.  How much heat per unit mass was added to the air parcel?

 

 

c.  How much work per unit mass was done in expanding the air parcel?

 

 

d.  What was the change in specific internal energy of the air parcel?

 

 

8.  A dry air parcel at an initial temperature of 20°C and a pressure of 950 mb is forced to rise adiabatically up a mountain slope.  The top of the mountain is at a pressure of 720 mb.

 

a.  What is the temperature of the air parcel when it reaches the top of the mountain?

 

 

 

b.  What is the work done by the air parcel?

 

 

 

9.  A cylinder filled with helium (a monatomic ideal gas) has a volume of 1.8x106 cm3, a pressure of 1.2x105 mb, and a temperature of 300K.  The cylinder is contained in an evacuated room with a volume of 16 m3.  The cylinder ruptures and helium fills the room.

 

a.  What is the pressure in the room after the cylinder ruptures?

 

 

b.  What is the temperature in the room after the cylinder ruptures?

 

 

c.  What is the work done by the expanding helium?

 

10.  A parcel of dry air is initially at a pressure of 900 mb and a temperature of 15°C.  It rises to the 400 mb level.

 

a.  What amount of heat (per mass) must be exchanged with its surroundings if the temperature is to remain constant at 15°C during the ascent?  Will the heat be gained or lost by the parcel?

 

b.  If the parcel first ascends adiabatically to 400 mb, and then heat is added to it to raise its temperature back to 15°C, how much heat must be added?  Is this the same amount of heat as the previous question?  If not, why not?

 

11.  Using the Poisson relation

show that

.