CK-12 People's Physics Book Version 2
Page 22
A sample of gas is used to drive a piston and do work. Here’s how it works: The gas starts out at standard atmospheric pressure and temperature. The lid of the gas container is locked by a pin.
The gas pressure is increased isochorically through a spigot to twice that of atmospheric pressure.
The locking pin is removed and the gas is allowed to expand isobarically to twice its volume, lifting up a weight. The spigot continues to add gas to the cylinder during this process to keep the pressure constant.
Once the expansion has finished, the spigot is released, the high-pressure gas is allowed to escape, and the sample settles back to .
Finally, the lid of the container is pushed back down. As the volume decreases, gas is allowed to escape through the spigot, maintaining a pressure of . At the end, the pin is locked again and the process restarts.
Draw the above steps on a diagram.
Calculate the highest and lowest temperatures of the gas.
A heat engine operates through cycles according to the diagram sketched below. Starting at the top left vertex they are labeled clockwise as follows: a, b, c, and d. From the work is and the change in internal energy is ; find the net heat.
From the a-c the change in internal energy is . Find the net heat from b-c.
From c-d the work is . Find the net heat from c-d-a.
Find the net work over the complete cycles.
The change in internal energy from b-c-d is . Find:
i. the net heat from c-d ii. the change in internal energy from d-a iii. the net heat from d-a
A sample mole of an ideal gas is taken from state A by an isochoric process to state B then to state C by an isobaric process. It goes from state C to D by a process that is linear on a diagram, and then it goes back to state A by an isobaric process. The volumes and pressures of the states are given below:
state Volume in Pressure in
A
B
C
D
Find the temperature of the states
Draw a diagram of the process
Find the work done in each of the four processes
Find the net work of the engine through a complete cycle
If of heat is exhausted in D-A and A-B and C-D are adiabatic, how much heat is inputted in B-C?
f. What is the efficiency of the engine?
Answers to Selected Problems
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molecules
a. b. Decreases to
c.
a. No b. allowed by highly improbable state. More likely states are more disordered.
a. b.
c.
d.
e. or
g.
a. b.
c.
a. % b.
c.
d.
e.
a. % b. %
c. %
b. c. isochoric; isobaric
d. to
e.
b.
a. b.
c.
d.
e.
Chapter 21: Gas Laws
In kinematics, once initial conditions of a system are set -- we are given the masses and positions of objects in question as well as the forces acting on them, we can theoretically obtain all future information about the system. By applying Newton's Laws, we can determine the positions and velocities of the objects at any point in time.
However, once we are discussing systems that consist of trillions of individual particles in constant motion, such a description becomes inadequate. In this case, instead of tracking the velocities and positions of each individual particle, we track several parameters, average quantities that sufficiently describe the system in question. Parameters typically defined in thermodynamics include pressure, denoted by the letter , volume (), and temperature ().
The concepts of pressure and volume should be familiar from previous sections, and the previous chapter discusses temperature.
Chapter 22: Heat Engines and The Laws of Thermodynamics
The Laws of Thermodynamics
Now that we have defined the terms that are important for an understanding of thermodynamics, we can state the laws that govern relevant behavior. These laws, unlike Newton's Laws or Gravity, are not based on new empirical observations: they can be derived based on statistics and known principles, such as conservation of energy. By understanding the laws of thermodynamics we can analyze heat engines, or machines that use heat energy to perform mechanical work.
The First Law
The First Law of Thermodynamics is simply a statement of energy conservation applied to thermodynamics systems: the change in the internal --- for our purposes, this is the same as thermal --- energy (denoted ) of a closed system is equal to the difference of net input heat and performed work. In other words, Note that this does not explain how the system will transform input heat to work, it simply enforces the energy balance.
The Second Law
The Second Law of Thermodynamics states that the entropy of an isolated system will always increase until it reaches some maximum value. Consider it in light of the simplified example in the entropy section: if we allow the low entropy system to evolve, it seems intuitive collisions will eventually somehow distribute the kinetic energy among the atoms.
The Second Law generalizes this intuition to all closed thermodynamic systems. It is based on the idea that in a closed system, energy will be randomly exchanged among constituent particles --- like in the simple example above --- until the distribution reaches some equilibrium (again, in any macroscopic system there will be an enormous number of of atoms, degrees of freedom, etc). Since energy is conserved in closed systems, this equilibrium has to preserve the original energy total. In this equilibrium, the Second Law --- fundamentally a probabilistic statement --- posits that the energy will be distributed in the most likely way possible. This typically means that energy will be distributed evenly across degrees of freedom.
This allows us to formulate the Second Law in another manner, specifically: heat will flow spontaneously from a high temperature region to a low temperature region, but not the other way. This is just applying the thermodynamic vocabulary to the logic of the above paragraph: in fact, this is the reason for the given definition of temperature. When two substances are put in thermal contact (that is, they can exchange thermal energy), heat will flow from the system at the higher temperature (because it has more energy in its degrees of freedom) to the system with lower temperature until their temperatures are the same.
When a single system is out equilibrium, there will be a net transfer of energy from one part of it to another. In equilibrium, energy is still exchanged among the atoms or molecules, but not on a system-wide scale. Therefore, entropy places a limit on how much work a system can perform: the higher the entropy, the more even the distribution of energy, the less energy available for transfer.
Heat Engines
Heat engines transform input heat into work in accordance with the laws of thermodynamics. The mechanics of various heat engines differ but their fundamentals are quite similar and involve the following steps:
Heat is supplied to the engine from some source at a higher temperature .
Some of this heat is transferred into mechanical energy through work done .
The rest of the input heat is transferred to some source at a lower temperature until the system is in its original state.
A single cycle of such an engine can be illustrated as follows:
In effect, such an engine allows us to 'siphon off' part of the heat flow between the heat source and the heat sink. The efficiency of such an engine is define as the ratio of net work performed to
input heat; this is the fraction of heat energy converted to mechanical energy by the engine:
If the engine does not lose energy to its surroundings (of course, all real engines do), then this efficiency can be rewritten as
A Carnot Engine, the most efficient heat engine possible, has an efficiency equal to
where and are the temperatures of the hot and cold reservoirs, respectively.
Application to Gases
The pressure of a gas is the force the gas exerts on a certain area. For a gas in a container, the amount of pressure is directly related to the number and intensity of atomic collisions on a container wall.
An ideal gas is a gas for which interactions between molecules are negligible, and for which the gas atoms or molecules themselves store no potential energy. For an “ideal” gas, the pressure, temperature, and volume are simply related by the ideal gas law.
Atmospheric pressure ( Pascals) is the pressure we feel at sea level due to the weight of the atmosphere above us. As we rise in elevation, there is less of an atmosphere to push down on us and thus less pressure.
When gas pressure-forces are used to move an object then work is done on the object by the expanding gas. Work can be done on the gas in order to compress it.
Adiabatic process: a process that occurs with no heat gain or loss to the system in question.
Isothermal: a process that occurs at constant temperature (i.e. the temperature does not change during the process).
Isobaric: a process that occurs at constant pressure.
Isochoric: a process that occurs at constant volume.
If you plot pressure on the vertical axis and volume on the horizontal axis, the work done in any complete cycle is the area enclosed by the graph. For a partial process, work is the area underneath the curve, or.
In a practical heat engine, the change in internal energy must be zero over a complete cycle. Therefore, over a complete cycle .
The work done by a gas during a portion of a cycle = , note can be positive or negative.
Key Equations
Temperature and kinetic energy:
The average kinetic energy of atoms (each of mass and average speed ) in a gas is proportional to the temperature of the gas, measured in Kelvin. This is just a restatement of the definition of temperature above. The Boltzmann constant is a constant of nature, equal to .
Definition of pressure:
The pressure on an object is equal to the force pushing on the object divided by the area over which the force is exerted. Unit for pressure are (called Pascals)
The Ideal Gas Law:
An ideal gas is a gas where the atoms are treated as point-particles and assumed to never collide or interact with each other. If you have molecules of such a gas at temperature and volume , the pressure can be calculated from this formula. Note that .
Different form of the Ideal Gas Law:
is the volume, is the number of moles; is the universal gas constant ; this is the most useful form of the gas law for thermodynamics.
Examples
Example 1
Question:A heat engine operates at a temperature of . The work output is used to drive a pile driver, which is a machine that picks things up and drops them. Heat is then exhausted into the atmosphere, which has a temperature of .
a) What is the ideal efficiency of this engine?
b.) The engine drives a weight by lifting it in . What is the engine’s power output?
c) If the engine is operating at of ideal efficiency, how much power is being consumed?
d) The fuel the engine uses is rated at . How many kg of fuel are used in one hour?
Answer:
a) We will plug the known values into the formula to get the ideal efficiency.
b) To find the power of the engine, we will use the power equation and plug in the known values.
c) First, we know that it is operating at of ideal efficiency. We also know that the max efficiency of this engine is . So the engine is actually operating at of efficiency. So is of what?
Chapter 23: BCTherm
Microscopic Description of an Ideal Gas
Evidence for the kinetic theory
Why does matter have the thermal properties it does? The basic answer must come from the fact that matter is made of atoms. How, then, do the atoms give rise to the bulk properties we observe? Gases, whose thermal properties are so simple, offer the best chance for us to construct a simple connection between the microscopic and macroscopic worlds.
A crucial observation is that although solids and liquids are nearly incompressible, gases can be compressed, as when we increase the amount of air in a car’s tire while hardly increasing its volume at all. This makes us suspect that the atoms in a solid are packed shoulder to shoulder, while a gas is mostly vacuum, with large spaces between molecules. Most liquids and solids have densities about 1000 times greater than most gases, so evidently each molecule in a gas is separated from its nearest neighbors by a space something like 10 times the size of the molecules themselves.
If gas molecules have nothing but empty space between them, why don’t the molecules in the room around you just fall to the floor? The only possible answer is that they are in rapid motion, continually rebounding from the walls, floor and ceiling. In chapter 2, we have already seen some of the evidence for the kinetic theory of heat, which states that heat is the kinetic energy of randomly moving molecules. This theory was proposed by Daniel Bernoulli in 1738, and met with considerable opposition, because there was no precedent for this kind of perpetual motion. No rubber ball, however elastic, rebounds from a wall with exactly as much energy as it originally had, nor do we ever observe a collision between balls in which none of the kinetic energy at all is converted to heat and sound. The analogy is a false one, however. A rubber ball consists of atoms, and when it is heated in a collision, the heat is a form of motion of those atoms. An individual molecule, however, cannot possess heat. Likewise sound is a form of bulk motion of molecules, so colliding molecules in a gas cannot convert their kinetic energy to sound. Molecules can indeed induce vibrations such as sound waves when they strike the walls of a container, but the vibrations of the walls are just as likely to impart energy to a gas molecule as to take energy from it. Indeed, this kind of exchange of energy is the mechanism by which the temperatures of the gas and its container become equilibrated.
Pressure, volume, and temperature
A gas exerts pressure on the walls of its container, and in the kinetic theory we interpret this apparently constant pressure as the averaged-out result of vast numbers of collisions occurring every second between the gas molecules and the walls. The empirical facts about gases can be summarized by the relation
PV nT, [ideal gas]
which really only holds exactly for an ideal gas. Here n is the number of molecules in the sample of gas.
The proportionality of volume to temperature at fixed pressure was the basis for our definition of temperature.
Pressure is proportional to temperature when volume is held constant. An example is the increase in pressure in a car’s tires when the car has been driven on the freeway for a while and the tires and air have become hot.
We now connect these empirical facts to the kinetic theory of a classical ideal gas. For simplicity, we assume that the gas is monoatomic (i.e., each molecule has only one atom), and that it is confined to a cubical box of volume V , with L being the length of each edge and A the area of any wall. An atom whose velocity has an x component vx will collide regularly with the left-hand wall, traveling a distance 2L parallel to the x axis between collisions with that wall. The time between collisions is ∆t = 2L/vx, and in each collision the x component of the atom’s momentum is reversed from -mvx to mvx. The total force on the wall is
F = + +...[monoatomic ideal gas] ,
where the indices 1, 2, . . . refer to the individual atoms. Substituting ∆px,i = 2mvx,i and ∆ti = 2L/vx,i, we have
F = + [monoatomic ideal gas] .
The quantity mvx,i2 is twice the contribution to the kinetic energy from the part of the atom’s center of mass motion that is parallel to the x axis. Since we’re assuming a monoatomic gas, center of mass motion is the only type of motion that gives rise to kinetic energy. (A more complex molecule could rotate and vibrate as well.) If the quantity inside the sum included the y and z components, it would be twice the total kinetic energy of all the molecules. By symmetry, it must therefore equal 2/3 of the total kinetic energy, so
F = [monoatomic ideal gas] .
Dividing by A and using AL = V , we have
P = [monoatomic ideal gas] .
This can be connected to the empirical relation PV nT if we multiply by V on both sides and rewrite KEtotal as nKEav, where KEav is the average kinetic energy per molecule:
PV = nKEav monoatomic ideal gas
For the first time we have an interpretation for the temperature based on a microscopic description of matter: in a monoatomic ideal gas, the temperature is a measure of the average kinetic energy per molecule. The proportionality between the two is KEav= (3/2)kT, where the constant of proportionality k, known as Boltzmann’s constant, has a numerical value of 1.38 x 10-23 J/K. In terms of Boltzmann’s constant, the relationship among the bulk quantities for an ideal gas becomes
PV = nkT , [ideal gas]
which is known as the ideal gas law. Although I won’t prove it here, this equation applies to all ideal gases, even though the derivation assumed a monoatomic ideal gas in a cubical box. (You may have seen it written elsewhere as PV = NRT, where N = U/NA is the number of moles of atoms, R = kNA, and NA = 6.0 X 1023, called Avogadro’s number, is essentially the number of hydrogen atoms in 1 g of hydrogen.)