Thermodynamics is the study of heat. Heat can be created, moved, stored, or used to do work. Heat is another kind of energy. Remember we said that energy was “stored work.”
You may think that heat and temperature are the same thing. The words get used interchangeably, so it can be confusing. Again, heat is a kind of energy. Temperature is what’s called a state variable. That means a measurement of an object’s state at a given time. Temperature is a way to determine how much heat an object may hold at that particular moment. Heat moves from warmer to colder so if a warm object is placed in contact with a cold object, the heat moves from the warmer object into the colder object.
Temperature is measured relative to a scale. There are many temperature scales; we’ll review the 3 that are most used.
The first is the Fahrenheit scale, most widely used in the US. There are several stories about how it was originally defined; the modern definition is relative to water. That is, the freezing point of water is 32F and the boiling point of water is 212F. That means there are 180 degrees between those two temperatures.
The second is a scale that has changed its name over the last 100 years. The Centigrade scale defined the freezing point of water as 0C, and the boiling point of water as 100C. Since there are 100 degrees between these two, it was called the Centigrade scale. Its name was later changed to the Celsius scale, still noted as C. Since these two scales (Fahrenheit and Celsius) have a different number of divisions for the same change in temperature (180 vs. 100), we see that the degrees are larger in the Celsius scale — almost twice as large ( to be exact).
The third scale we will mention is the Kelvin scale. It is referenced to absolute zero, a temperature so cold that all motion stops, even molecules and electrons. This zero point, which is about C, is called 0 K. Note that we don’t use a degree symbol with the Kelvin scale. It uses degrees that are the same size as Celsius degrees, so the freezing point of water is 273.15 K and the boiling point of water is 373.15 K.
The Laws of Thermodynamics
The study of Thermodynamics is based on “Laws.” These laws are not derived, they are simply stated and accepted. We will find other such laws as we move through other parts of physics. There are three laws of thermodynamics that are generally accepted. Others have added an additional law, and called it the zeroth law of thermodynamics. Again, we don’t try to prove these laws, we accept them and build the rest of our understanding from them.
Zeroth Law of Thermodynamics
This newest law of thermodynamics is a statement about two objects in thermal equilibrium with each other. Thermal equilibrium means that the two objects in contact with each other are at the same temperature, so no heat flows between them. That’s what it means to be in equilibrium.
So this law says: If object A is in thermal equilibrium with object B, and object C is in thermal equilibrium with object B, then object A and object C are in thermal equilibrium with each other. The mathematical equivalent is: If A = B and B = C, then A = C.
The First Law of Thermodynamics
This law is about the conservation of energy. In an isolated system, energy cannot be created nor destroyed. In thermodynamics, we define systems. It’s often a heat loop, or a cycle. If energy leaves the system (or loop), then the first law of thermodynamics says that the total energy of the system must go down. In other words energy must be conserved.
The Second Law of Thermodynamics
This law discusses a new quantity, called entropy. Entropy is a concept in thermodynamics that describes the amount of chaos or disorder in a system. When left alone, the entropy in a system cannot decrease (think of a teenager’s bedroom). It takes work to increase the order of the system, which decreases the entropy. The second law of Thermodynamics says that if no work is being done on a system, then its total entropy can never decrease.
The Third Law of Thermodynamics
The entropy of a system approaches a constant value as the temperature approaches absolute zero. So getting a system close to absolute zero causes its entropy to stop increasing (become constant). We won’t have too much to say about this law — I put it here for completeness.
Perpetual Motion Machines
Perpetual motion machines are theoretical devices that do not exist. These machines can be divided into two kinds. The first kind is a machine that can produce work with no energy input. The first law of thermodynamics prohibits such a machine.
Perpetual motion machines of the second kind are machines that can convert thermal energy directly into work. What does this mean? All machines built so far use a medium of transfer for converting heat into work. For example a steam turbine uses water — the water is heated, turning it into steam that expands and pushes a piston. It is not a direct conversion of heat into work. So, the second law of Thermodynamics prohibits these machines.
Most materials, when heated, expand. First, let’s talk about a bar of material — something that is long in one direction. We can deal with the expansion in the long direction first. When we are limiting the expansion to one dimension, we call that linear expansion. So, the change in length () can be described by the equation
where is the expansion rate of the material that the bar is made from, and is the change in length, and
Next, we talk about expansion in three dimensions, called volume expansion. Since we have 3 directions expanding at the same time and same rate, we write it like so
where So the volume expansion coefficient is the linear expansion coefficient — because it expands in the and directions equally.
The heat capacity of a given mass of a specific material is given by
where is the heat capacity, is the mass and is called the specific heat of the material, and is the change in temperature.
So, with this relationship (equation), we can calculate the amount of heat (energy) needed to raise the temperature of a substance, or, given a temperature change in a substance, calculate the heat required.
So let’s work an example. Say we wanted to heat one liter (1 l = 1 kg) of water from C to C, so here the change in temperature, C. How much heat (energy) would be required?
First, we look up the specific heat of water, and find that it is Then, we can directly calculate
It takes a lot of energy to heat water!
So this is how you can calculate the heat needed to raise the temperature of a given amount of some material.
Matter exists in four phases: solid, liquid, gas, and plasma. Here, we will discuss the first 3. Let’s use water as an easy-to-understand example. Water in the solid phase is called ice, liquid phase is called water, and gas phase is called water vapor. As a substance move from one phase to another, this is called a phase transition.
So, to calculate the the heat needed to raise the temperature across a phase transition, we need to add (as appropriate) for solid-liquid transition (melting) the latent heat of fusion, and for liquid-gas transition (boiling), the latent heat of vaporization. As these phase-change boundaries are crossed, the temperature stays constant. That’s why when heat is added to ice and it starts melting, the temperature does not increase until all the ice is melted.
Again, let’s work an example. Say we had the same 1 kg of mass, but it starts as ice at C, and we want to take it to water vapor at C. How much heat is needed?
First, we need to look up the latent heat values. For water, the latent heat of fusion is kJ/kg and the latent heat of vaporization is kJ/kg. So, to do this calculation, we need to heat up the material to each phase transition, then add the latent heats. We’ll do this in pieces so it’s easier to understand.
So, in words, we start at C, heat to C, add latent heat of fusion (melt it), heat to C, add latent heat of vaporization (boil it), then heat to C.
So this calculation looks like:
Then, substituting numbers, we get
How Can Heat do Work?
Recall from earlier that the definition of work is force () along a displacement (), written as
Another way to think of this is: incremental (additional) work is force times incremental displacement, written as
Remember that the “” means this is “a very small change in,” also known as an infinitesimal.
Let’s look at a piston in a cylinder that pushes on or is pushed by the gas in the cylinder. The gas is pushing down or the piston is pushing up; the displacement is also up or down, so we’ll define it as the direction, and the infinitesimal displacement, we’ll call giving
Since the area of the cylinder does not change (call it ), when the piston moves an infinitesimal amount (), then the infinitesimal change in volume above the piston can be written as
Next, we define a term, pressure, something you should be familiar with. What it means is force per area. So the pressure of the gas in this cylinder is the force it exerts () on the piston, divided by the area of the piston, We write pressure like so
Re-arranging, we get Next, we can substitute this into the work equation, giving
Re-grouping, we get
Finally, to calculate the work, we need to “add up” all the infinitesimals by calculating the integral of this equation. Since any variable raised to the 0th power is just 1, we can say that is the same as Remember from chapter 3 that the “recipe” for an integral is to increase the power by one and divide by the new power, so
Now, integrate both sides of the work equation using limits of integration starting with initial volume and going to final volume giving
So the left side gives and since pressure is essentially constant over a very small movement of the piston, we can treat pressure as a constant and move it in front of the integral, like so
again here, and since we have limits of integration, we write it like so
to use the limits of integration, we evaluate the result of the integral at upper limit minus the result at the lower limit, giving
This tells us that work equals pressure times the change in volume. A common thing to do in thermodynamics is to plot pressure vs. volume; this is called a PV curve. So, since the definition of an integral is area under the curve, so we can also say that work is the area under the PV curve. Next we’ll look at some PV curves and interpret what they’re telling us.
PV Diagrams and Work
As we just said, work is equal to the area under the curve on a PV diagram. But the question we now have to answer is: Is the work positive or negative? What we mean by that is: Is the system producing work or is it consuming work? Is the gas in the cylinder being worked on or is it producing work by pushing back on the piston. By system, we mean the gas in the cylinder. So if we’re doing work on the gas i.e., pushing on the piston, then we are doing work on the gas, which means the gas is consuming the work and the sign is negative. But the other way around, if the gas is pushing on the piston then the system is producing work, which is positive.
In thermodynamics we talk about cycles; if we move from one point on the PV diagram to a different point on the PV diagram and then return, this is called a cycle. So if work is consumed in one direction and produced in the other direction, as long as positive is greater than negative, then the cycle can produce work. Let’s look at an example.
In diagram (a) we’re going from one volume to a smaller volume and from one pressure to a higher pressure; that means the piston is compressing the gas. So we are doing work on the gas and the system is consuming work so in this case work is negative. In diagram (b) we go from a high pressure to a low pressure and at the same time as we go from a smaller volume to a bigger volume. This means that the gas is pushing out on the piston, so the gas is producing work, and in this case the work is positive.
In diagram (c) we subtract one from the other. Again work is the area under the curve. So since the positive work has more area under the curve in the negative work, when we subtract two, we have positive work left over. That means that the cycle shown here is capable of producing work.
Is this the only answer? No, Figure (d) shows that if we hold the volume constant and reduce the pressure then hold the pressure constant and increase the volume, we transition between the same two points as before, but less work is done, because there is less area under the curve. So the amount of work done is completely dependent upon the path that the system takes from one point to another. Since work is the area under the curve, the amount of work done is completely dependent upon the path taken from one point to another point on the PV diagram.
Studies of these diagrams and many different cycles, some much more complicated and showing transitions between more than two points is a whole field of study. This is typically taught in an entire semester in a college engineering program and is called Thermodynamics I.
In the next chapter we will study something called the Ideal Gas Law, and we’ll include temperature in the mix, in addition to pressure and volume.
(a) Piston doing work on the gas, so
(b) Gas doing work on the piston, so
(c) Full cycle — difference of (b) – (a);
(d) Same endpoints as (b), but less work is done (the area under the curve is smaller)
Figure: Example PV diagrams, showing positive and negative
The First Law of Thermodynamics
Imagine an isolated system that contains a quantity of heat energy that we’ll call and the amount of work it does (or has done to it) we’ll call Then we can write a relationship for the internal energy of the system like so
which is an equation that represents the first law of thermodynamics. What this tells us is that any change in either or will cause a change in the internal energy of the system. This law is equivalent to the conservation of energy that we saw back in chapter 10.
There are some special cases to describe and define their names.
- If no heat is transferred into or out of the system, then This type of process is called adiabatic. In the first law equation is
- If we look back at frame (d) in the diagram figure, the transition from the initial state to the final state is in two straight line segments. The first one, which moves straight down, indicates a process in which pressure changes, but volume does not. This is called an isovolumetric process. The second straight line segment is horizontal, and here volume changes but pressure does not. This is called an isobaric process.
- If we have a cyclic process, where the system comes back around to the initial state, then there is no change in internal energy, so and the first law equation tells us that This means that the amount of heat transferred into the system equals the amount of work done.
Suppose you had gas in a cylinder, and you did of work on the piston. You are told that the system absorbed of heat energy. How much did the total internal energy change?
Since the piston is doing work, then the work done on the system is The system increased its heat energy by , so the first law tells us
Heat Transfer Mechanisms
Heat can be transferred from one object to another in three ways. If the two objects are in direct contact with each other, then heat is transferred between them by conduction. If two objects are not touching each other but there is air flowing past them, then heat is transferred by the air; this is called convection. The third heat transfer mechanism is called radiation. When you feel heat from a hot object like a stove or a campfire at a distance, this is radiant heat. It does not require being in physical contact to transfer heat nor does it require the presence of air. This is how the sun heats the Earth, and how satellites in orbit can remove excess heat from their electronics — by radiating it into deep space.
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