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# Temperature

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Students will learn the concept of temperature, where it comes from and how it relates to our everyday world.

### Key Equations

$KE_{avg} = \frac{1}{2} m v ^2_{avg} = \frac{3}{2}kT$

The average kinetic energy of atoms (each of mass $m$ and average speed $v$ ) in a gas is related to the temperature $T$ of the gas, measured in Kelvin. The Boltzmann constant $k$ is a constant of nature, equal to $1.38\times10^{-23} \;\mathrm{J/K}$

$T_F = \frac{9}{5}T_C + 32^\circ \text{F}$ ; conversion from Celsius to Fahrenheit

$T_C = \frac{5}{9}(T_F - 32^\circ \text{F})$ ; conversion from Fahrenheit to Celsius

### Guidance

• When an object feels cold to the touch, it is because heat is flowing from you to the object.
• When an object feels hot to the touch, it is because heat is flowing from the object to you.
• Some objects (like metals) conduct heat better than others (like wood). Thus if you stick a metal rod in the fireplace and hold the other end, the heat is conducted well and you get burned. On the other hand, if you place a wood stick in the fire and hold the other end you’ll be OK.
• The temperature of a gas is a measure of the amount of average kinetic energy that the atoms in the gas possess.
• If you heat something, you increase its internal energy, so you increase the movement of molecules that make up this thing, thus it expands. This is called heat expansion; most everything expands as heated and contracts as cooled.
• Most materials expand as they are heated. This can cause bridges to collapse if they are not designed to have a place to expand in the summer months (like the placing of metal ‘teeth’ at intervals on the Golden Gate Bridge).
• Water contracts from $0^\circ C$ to $4^\circ C$ and then expands from $4^\circ C$ to $100^\circ C$ . Remembering that density is mass divided by volume explains why water at $4^\circ C$ is more dense than water below and above $4^\circ C$ . This also explains why lakes freeze on the top first and not throughout. As the water on the top of the lake drops below $4^\circ C$ , it is now more dense than the water below it, thus it sinks to the bottom, allowing the warmer water to rise up to the top and cool down in the winter weather. Only when the entirety of the lake is at $4^\circ C$ , then the lake can start to freeze. It freezes from the top down, because water below $4^\circ C$ is less dense than water at $4^\circ C$ .

• More than 50% of the water rise expected from global warming is due to the thermal expansion of water
• There are 3 different temperature scales you should know-the Kelvin scale, the Celsius scale and the Fahrenheit scale.
• The Kelvin scale (K) is the one used in most scientific equations and has its zero value set at absolute zero (the theoretical point at which all motion stops).
• The Celsius scale $(^\circ C)$ is the standard SI temperature scale. It is equal to the Kelvin scale if you minus 273 from the Celsius reading. Water has a boiling point of $100^\circ C$ and a freezing point of $0^\circ C$ .
• The Fahrenheit scale $(^\circ F)$ is the English system and the one we are familiar with.
• Newtons’ Law of Cooling: The rate of heat transfer is proportional to the difference in temperature between the two objects. For example, hot liquid that is put in the freezer will cool much faster than a room temperature liquid that is put in the same freezer.

#### Kinetic Theory of Gases

According to classical kinetic theory, temperature is always proportional to the average kinetic energy of molecules in a substance. The constant of proportionality, however, is not always the same.

Consider: the only way to increase the kinetic energies of the atoms in a mono-atomic gas is to increase their translational velocities. Accordingly, we assume that the kinetic energies of such atoms are stored equally in the three components ( $x,y, \text{ and } z$ ) of their velocities.

On the other hand, other gases --- called diatomic --- consist of two atoms held by a bond. This bond can be modeled as a spring, and the two atoms and bond together as a harmonic oscillator. Now, a single molecule's kinetic energy can be increased either by increasing its speed, by making it vibrate in simple harmonic motion, or by making it rotate around its center of mass . This difference is understood in physics through the concept of degrees of freedom : each degree of freedom for a molecule or particle corresponds to a possibility of increasing its kinetic energy independently of the kinetic energy in other degrees.

It might seem that monatomic gases should have one degree of freedom: their velocity. They have three because 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 . Let's try to understand why this is in reference to our monoatomic ideal gas. In the derivation above, volume was constant; so, temperature was essentially proportional to pressure, which in turn was proportional to the kinetic energy due to translational motion of the molecules. If the molecules had been able to rotate as well as move around the box, they could have had the same kinetic energy with slower translational velocities, and, therefore, lower temperature. In other words, in that case, or assumption that the kinetic energy of the atoms only depends on their velocities, implied between equations [2] and [3], would not have held . Therefore, the number of degrees of freedom in a substance determines the proportionality between molecular kinetic energy and temperature: the more degrees of freedom, the more difficult it will be to raise its temperature with a given energy input. This is why it takes so long to boil water but so little time to heat up a piece of metal with the same mass.

A note about the above discussion:
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. Unlike most cases in your current physics class, where these can be ignored, in this case, quantum effects can make an appreciable difference. For instance, the vibrational degrees of freedom of diatomic gas molecules discussed above are, for many gases, inaccessible in very common conditions, although we do not have the means to explain this within our theory. In fact, this was one of the first major failures of classical physics that ushered in the revolutionary discoveries of the early 20th century.

#### Example 1

Convert 75 degrees Fahrenheit to Celcius.

##### Solution

To do this conversion, we'll just use the equation given above.

$T_c&=\frac{5}{9}(T_f - 32^\circ \text{F})\\T_c&=\frac{5}{9}(75^\circ \text{F} - 32^\circ \text{F})\\T_c&=23.8^\circ \text{C}$

#### Example 2

The mass of a neon atom is $3.34*10^{-26}\;\text{kg}$ . If the temperature of the neon atom 100 K, what is it's average velocity?

##### Solution

We can solve this problem using the equation given above relating kinetic energy to the temperature of a gas.

$\frac{1}{2}mv^2&=\frac{3}{2}kT\\v&=\sqrt{\frac{3kT}{m}}\\v&=\sqrt{\frac{3*1.38*10^{-23}\;\text{J/K} * 100\;\text{K}}{3.34*10^{-26}\;\text{kg}}}\\v&=352.1\;\text{m/s}\\$

### Time for Practice

1. Convert the boiling point of water $100^\circ \text{C}$ to Fahrenheit
2. Convert $70^\circ \text{F}$ to Celsius
3. 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.
4. Explain why a lake freezes at the top first, rather than throughout or at the bottom?
5. Typical room temperature is about $300 \;\mathrm{K}$ . As you know, the air in the room contains both $O_2$ and $N_2$ 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?
6. Temperature is a measure of the average kinetic energy of molecules. Consider a mixture of hydrogen and oxygen gas at a certain temperature. Use the formula for kinetic energy to explain why the hydrogen molecules move faster than the oxygen molecules. Then explain why there’s no hydrogen in the earth’s atmosphere, and why the moon has no atmosphere at all.
7. Calculate the average speed of $N_2$ molecules at room temperature $(300 \;\mathrm{K})$ . (You remember from your chemistry class how to calculate the mass (in $kg$ ) of an $N_2$ molecule, right?)
8. How high would the temperature of a sample of $O_2$ gas molecules have to be so that the average speed of the molecules would be $10$ % the speed of light? Convert this temperature to Kelvin and Fahrenheit as well.

1. $212^\circ \text{F}$
2. $21^\circ \text{C}$
3. .
4. .
5. .
6. .
7. $517 \;\mathrm{m/s}$
8. $1.15 \times 10^{12}\;\mathrm{K}$