It would seem intuitive that atoms in monatomic gases have one degree of freedom: their velocity. In fact, they actually have three, since their velocity can be altered in one of three mutually perpendicular directions without changing the kinetic energy in other two --- just like a centripetal force does not change the kinetic energy of an object, since it is always perpendicular to its velocity. These are called translational degrees of freedom.
Diatomic gas molecules, on the other hand have more: the three translational explained above still exist, but there are now also vibrational and rotational degrees of freedom. Monatomic and diatomic degrees of freedom can be illustrated like this:
Temperature is an average of kinetic energy over degrees of freedom, not a sum. Therefore, generally, the more degrees of freedom, the more difficult it will be to raise its temperature with a given amount of heat flow. This explanation is not quite complete: since the objects at the basis of our understanding of thermodynamics are atoms and molecules, quantum effects can make certain degrees of freedom inaccessible at specific temperature ranges -- in fact, this was one of the first major failures of classical physics that ushered in the revolutionary discoveries of the early 20th century.
Measuring Temperature
Temperature Scales
There are different temperature scales: defining the units of this measure is up to us. The most frequently used ones are called Fahrenheit, Celsius, and Kelvin, after their respective popularizers.
So, does any particular choice of units for temperature matter? Not really; any consistent scale will work. Consider the SI unit of length, the meter: it was actually originally defined as one ten-millionth of the distance from the Equator to the North Pole through Paris. This choice was entirely arbitrary in any universal sense, but it served physics completely adequately. Any other unit of distance would have worked as well.
The slightly tricky aspect to scales of measure is that --- to completely define one --- in addition to picking the unit size in terms of physical quantities (which is clear from above) one needs to set a zero level. When dealing with length scales, the zero level is apparent: the complete absence of length. Because of this, 0 miles is the same as 0 feet and 0 meters and conversion between lengths is a matter of multiplication alone. This seems obvious, but let's ask a more subtle question: would a length scale where a length of zero corresponds to, say 1 meter on the SI scale, work consistently? The answer is that yes, it would, but it would be cumbersome in mathematical analysis; there would be negative lengths: a rather counter-intuitive concept.
The definition of temperature above suggests an obvious zero for temperature as well: when the average kinetic energy of molecules in a substance is zero, or all motion has stopped. If all our scales were set up like this, conversion between temperature scales would be as easy as between length scales: simple multiplication (miles to kilometers, meters to centimeters, etc). Unfortunately, the Celsius and Fahrenheit scales were created before temperature was this well defined, so they assign the value of '0 degrees' to arbitrary points, and, therefore, have negative temperatures and are cumbersome in mathematical analysis.
The Celsius scale --- used throughout most of the world --- establishes its unit, or degree Celsius, by defining the temperature difference between the freezing and boiling point of water as 100 degrees Celsius. This is analogous to the definition of meter above. However, it assigns a temperature of 0 to the freezing point of water; this temperature is considerably higher than absolute zero.
Thus, scientists generally use the Kelvin temperature scale, which has degree increments equal to the Celsius scale's (and so is pretty easy to recognize and interpret --- at least for people outside the U.S.), but sets the value of zero temperature to the absolute zero --- the point at which all molecular motion ceases. On the Celsius scale, this temperature is -273.15, so to convert between from degrees Celsius to Kelvins (frustratingly, while we call the Celsius Scale units degrees Celsius, the Kelvin scale units are conventionally referred to simply as Kelvins) we us the following:
For the rest of this chapter, temperature will is assumed to be measured in Kelvins.
Specific Heat Capacity and Specific Latent Heat
The ideas in the paragraphs above can be understood better through the concept of specific heat capacity (or specific heat for short), which relates an increase in temperature of some mass of a substance to the amount of heat required to achieve it. In other words, for any substance, it relates thermal energy transfers to changes in temperature. It has units of Joules per kilogram Kelvin. Here is how we can define and apply specific heat ( refers to heat supplied, to the mass of the substance and to its specific heat capacity):
Heat capacity is largely determined by the number of degrees of freedom of the molecules in a substance. However, it also depends on other parameters, such as pressure. Therefore, the formula above implicitly assumes that these external parameters are held constant (otherwise we wouldn't know if we're measuring a change in specific heat is real or due to a change in pressure).
When a substance undergoes a phase change, its temperature does not change as it absorbs heat. We referred to this as an increase or decrease in latent energy earlier. In this case, the relevant question is how much heat energy does it require to change a unit mass of the substance from one phase to another? This ratio is known as latent heat, and is related to heat by the following equation ( refers to the latent heat):
During a phase change, the number of degrees of freedom changes, and so does the specific heat capacity. Heat capacity can also depend on temperature within a given phase, but many substances, under constant pressure, exhibit a constant specific heat over a wide range of temperatures. For instance, here is a graph of temperature vs heat input for a mole ( molecules) of water. Note that the x-axis of the graph is called 'relative heat energy' because it takes a mole of water at 0 degrees Celcius as the reference point.
The sloped segments on the graph represent increases in temperature. These are governed by equation [1]. The flat segments represent phase transitions, governed by equation [2]. Notice that the sloped segments have constant, though different, slopes. According to equation [1], the heat capacity at any particular phase would be the slope of the segment that corresponds to that phase on the graph. The fact that the slopes are constant means that, within a particular phase, the heat capacity does not change significantly as a function of temperature.
Entropy
The last major concept we are going to introduce in this chapter is entropy. We noted earlier that temperature is determined not just by how much thermal energy is present in a substance, but also how it can be distributed. Substances whose molecules have more degrees of freedom will generally require more thermal energy for an equal temperature increase than those whose molecules have fewer degrees of freedom.
Entropy is very much related to this idea: it quantifies how the energy actually is distributed among the degrees of freedom available. In other words, it is a measure of disorder in a system. An example may illustrate this point. Consider a monatomic gas with atoms (for any appreciable amount of gas, this number will be astronomical). It has degrees of freedom. For any given value of thermal energy, there is a plethora of ways to distribute the energy among these. At one extreme, it could all be concentrated in the kinetic energy of a single atom. On the other, it could be distributed among them all. According to the discussion so far, these systems would have the same temperature and thermal energy. Clearly, they are not identical, however. This difference is quantified by entropy: the more evenly distributed the energy, the higher the entropy of the system. Here is an illustration:
Thermodynamics and Heat Engines Problem Set
Consider a molecule in a closed box. If the molecule collides with the side of the box, how is the force exerted by the molecule on the box related to the momentum of the molecule? Explain conceptually, in words rather than with equations.
If the number of molecules is increased,
how is the pressure on a particular area of the box affected? Explain conceptually, in words rather than with equations.
The temperature of the box is related to the average speed of the molecules. Use momentum principles to relate temperature to pressure. Explain conceptually, in words rather than with equations.
What would happen to the number of collisions if temperature and the number of molecules remained fixed, but the volume of the box increased? Explain conceptually, in words rather than with equations.
Use the reasoning in the previous four questions to qualitatively derive the ideal gas law.
Typical room temperature is about . As you know, the air in the room contains both and gases, with nitrogen the lower mass of the two. If the average kinetic energies of the oxygen and nitrogen gases are the same (since they are at the same temperature), which gas has a higher average speed?
Use the formula to argue why it is easier to pop a balloon with a needle than with a finger (pretend you don’t have long fingernails).
Take an empty plastic water bottle and suck all the air out of it with your mouth. The bottle crumples. Why, exactly, does it do this?
You will notice that if you buy a large drink in a plastic cup, there will often be a small hole in the top of the cup, in addition to the hole that your straw fits through. Why is this small hole necessary for drinking?
Suppose you were swimming in a lake of liquid water on a planet with a lower gravitational constant than Earth. Would the pressure meters under the surface be the same, higher, or lower, than for the equivalent depth under water on Earth? (You may assume that the density of the water is the same as for Earth.)
Why is it a good idea for Noreen to open her bag of chips before she drives to the top of a high mountain?
Explain, using basic physics conservation laws, why the following conditions would cause the ideal gas law to be violated: There are strong intermolecular forces in the gas.
The collisions between molecules in the gas are inelastic.
The molecules are not spherical and can spin about their axes.
The molecules have non-zero volume.
To the right is a graph of the pressure and volume of a gas in a container that has an adjustable volume. The lid of the container can be raised or lowered, and various manipulations of the container change the properties of the gas within. The points and represent different stages of the gas as the container undergoes changes (for instance, the lid is raised or lowered, heat is added or taken away, etc.) The arrows represent the flow of time. Use the graph to answer the following questions.
Consider the change the gas undergoes as it transitions from point to point . What type of process is this? adiabatic
isothermal
isobaric
isochoric
entropic
Consider the change the gas undergoes as it transitions from point to point . What type of process is this? adiabatic
isothermal
isobaric
isochoric
none of the above
Consider the change the gas undergoes as it transitions from point to point . Which of the following best describes the type of process shown? isothermal
isobaric
isochoric
How would an isothermal process be graphed on diagram?
Write a scenario for what you would do to the container to make the gas within undergo the cycle described above.
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Calculate the average speed of molecules at room temperature . (You remember from your chemistry class how to calculate the mass (in ) of an molecule, right?)
How high would the temperature of a sample of gas molecules have to be so that the average speed of the molecules would be % the speed of light?
How much pressure are you exerting on the floor when you stand on one foot? (You will need to estimate the area of your foot in square meters.)
Calculate the amount of force exerted on a patch of your skin due to atmospheric pressure . Why doesn’t your skin burst under this force?
Use the ideal gas law to estimate the number of gas molecules that fit in a typical classroom.
Assuming that the pressure of the atmosphere decreases exponentially as you rise in elevation according to the formula , where is the atmospheric pressure at sea level , is the altitude in km and a is the scale height of the atmosphere . Use this formula to determine the change in pressure as you go from San Francisco to Lake Tahoe, which is at an elevation approximately above sea level.
If you rise to half the scale height of Earth’s atmosphere, by how much does the pressure decrease?
If the pressure is half as much as on sea level, what is your elevation?
At Noah’s Ark University the following experiment was conducted by a professor of Intelligent Design (formerly Creation Science). A rock was dropped from the roof of the Creation Science lab and, with expensive equipment, was observed to gain of internal energy. Dr. Dumb explained to his students that the law of conservation of energy required that if he put of heat into the rock, the rock would then rise to the top of the building. When this did not occur, the professor declared the law of conservation of energy invalid. Was the law of conservation of energy violated in this experiment, as was suggested? Explain.
If the law wasn’t violated, then why didn’t the rock rise?
An instructor has an ideal monatomic helium gas sample in a closed container with a volume of , a temperature of , and a pressure of . Approximately how many gas atoms are there in the container?
Calculate the mass of the individual gas atoms.
Calculate the speed of a typical gas atom in the container.
The container is heated to . What is the new gas pressure?
While keeping the sample at constant temperature, enough gas is allowed to escape to decrease the pressure by half. How many gas atoms are there now?
f. Is this number half the number from part (a)? Why or why not? g. The closed container is now compressed isothermally so that the pressure rises to its original pressure. What is the new volume of the container? h. Sketch this process on a P-V diagram. i. Sketch cubes with volumes corresponding to the old and new volumes.
A famous and picturesque dam, high, releases of water a second. The water turns a turbine that generates electricity. What is the dam’s maximum power output? Assume that all the gravitational potential energy of the water is converted into electrical energy.
If the turbine only operates at % efficiency, what is the power output?
How many Joules of heat are exhausted into the atmosphere due to the plant’s inefficiency?
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 . What is the ideal efficiency of this engine?
The engine drives a weight by lifting it in . What is the engine’s power output?
If the engine is operating at % of ideal efficiency, how much power is being consumed?
How much power is exhausted?
The fuel the engine uses is rated at . How many kg of fuel are used in one hour?
Calculate the ideal efficiencies of the following sci-fi heat engines: A nuclear power plant on the moon. The ambient temperature on the moon is . Heat input from radioactive decay heats the working steam to a temperature of .
A heat exchanger in a secret underground lake. The exchanger operates between the bottom of a lake, where the temperature is , and the top, where the temperature is .
A refrigerator in your dorm room at Mars University. The interior temperature is ; the back of the fridge heats up to .
How much external work can be done by a gas when it expands from to in volume under a constant pressure of ? Can you give a practical example of such work?
In the above problem, recalculate the work done if the pressure linearly decreases from to
under the same expansion. Hint: use a diagram and find the area under the line.
One mole of an ideal gas is moved through the following states as part of a heat engine. The engine moves from state A to state B to state C, and then back again.
State Volume Pressure Temperature
A
B
C
Draw a P-V diagram.
Determine the temperatures in states A, B, and C and then fill out the table.
Determine the type of process the system undergoes when transitioning from A to B and from B to C. (That is, decide for each if it is isobaric, isochoric, isothermal, or adiabatic.)
During which transitions, if any, is the gas doing work on the outside world? During which transitions, if any, is work being done on the gas?
What is the amount of net work being done by this gas?
CK-12 People's Physics Book Version 2 Page 21