# Gases

Molecules in gases

Gases contain molecules floating around and bumping into each other. Take air for example. Air contains mainly molecules of Nitrogen, Oxygen, water vapor, Argon, and Carbon Dioxide. At the molecular lengthscale, these different molecules are bumping into each other constantly; like bumping cars at the arcade. But why do molecules in gases keeping bumping around and moving?

Molecules keep bumping around because of temperature. The higher the temperature, more the faster the molecules move and bump into each other. Molecules at a high temperature are like toddlers in a small room; just after they have eaten loads of candy. They have a lot of energy and keep randomly whizzing around. When the temperature becomes lower, molecules move slower similar to toddlers who have already run around for half an hour and are now tired.

# Absolute Zero

You might ask if molecules get slower with decreasing temperature, when do they completely cease motion; similar to tired out toddlers taking a nap? (I apologise for the toddler analogies - that's the only other thing on my mind these days). It turns out that temperature is a very specific one, called absolute zero. Absolute zero is the coldest temperature ever! It's -2730 C or -459F. (See the image for temperature conversions.) Absolute zero sets the temperature scale most commonly used by scientists, called the Kelvin scale. In this scale, 0K corresponds to -2730C. Converting to Kelvin is an important first step in solving problems relating to gases that we will do later.

The figure below shows the actual distribution of molecular speeds at various temperatures for a certain gas. What trends do you see?

What are the properties of a gas? When your AC blows air, it fills up an enclosed space. Volume is one of the key properties of a gas. What about when you fill air in your car's tires? You fill it to a certain pressure. Pressure is another key property. The temperature is the third important property. Finally, number of molecules is the last important property. As air fills up your car's tires the number of air molecules increases. Pressure, volume and temperature and number are the key properties of a gas. Turns out they are related to each other through the ideal gas law. But before we go into the ideal gas law, let's look closely at each of these factors.

# Volume

Volume is how much space an object can occupy. Seems like a simple concept. However, I've seen many students struggling when doing math related to volume vs area vs length, which are all related but different concepts. So how is volume different from area? You might remember that length is measured in meters (m); area is measured in squared meters (m2); and volume is measured in cubic meters (m3). An alternate measure of dimension is centimeters (cm) or inches (in) that is commonly used across the US. You might also remember that 2.54 cm make an inch. However, the problem arises when I ask how to convert area in (m2) to cm2 or god forbid in2. This is all very important to be familiar with. Think about if you are buying a house and Redfin tells you the size in acres, and you want to get a sense of how big it is in (m2), and how long in meters it is to walk from one end of the property to the other. Now let's talk about these concepts through concrete examples.

Let's say your classroom is 6 m long. One day, your school is overrun by cow activists, and cows decide to "occupy the classroom". The length of these cows is around 2m. Assuming cows line up the back wall in your classroom, how many cows can fit against a 6m wall? Well the answer to this is 3 cows, (6m/2m=3). This is a length problem. However, now let's say a month later there are many more cows that join their activist friends. Now, the cows decide to completely fill the classroom. For the sake of simplicity let's assume "square cows" that are 2m x 2m. In this case, how many cows can fit in the classroom? Well now it's 3x3 = 9 cows.

Finally, consider the (even more) absurd scenario where the cows are really mad, and decide to completely occupy the class space. How so? They pile up on each other as much as the class will allow. And you guessed right; let's assume 'cubic cows' of dimension 2mx2mx2m! Now how many cows can fit in the class? It is 3x3x3= 27 cows. That's a whole lot more than just the 3 cows standing at the back wall!

Why all this talk about cows? You might think it fairly obvious. The common mistake I've seen while converting between units is many students are aware that 100 cm =1 m. However, it does not mean that the units of area have the same conversion, i.e. 100 cm2 is not 1 m2, and 100 cm3 is not 1 m3. Saying that 100 cm2 equals 1 m2 is equivalent to the statement that only 3 cows can fill an entire classroom. Similar to the cow analogy, 100x100 cm2 = 1 m2 and 100x100x100 cm3 = 1 m3. Another way to think about it is in a length of 1m you can fit 100 cm, in an area of 1 m2 you can fit 100x100=10,000 cm sized squares, and in a volume of 1 m3 you can fit 100x100x100=1,000,000 cm sized cubes.

# Pressure

A common misconcept is that pressure is the same as force. Yes it's related to force but it's more. Pressure is Force/ Area and that is an important difference. Consider laying on a bed of nails vs standing on the same bed. In both cases the force is the same, arising from gravity. However, standing on the bed of nails is a lot more pressure. A simple calculation shows that pressure from standing on a bed of nails can be nearly 50 times higher, and that's what causes your skin to break. For this calculation you need to first estimate force of gravity. Remember that force (measured in Newtons, or N for short) is Mass (measured in kg) times gravity (a constant value at sea level of 9.8 m/s2). If force is measured in N and area in m2, resulting pressure is in the units of Pascals (Pa).

# Energy

While we learnt about forces in Chapter 2., a lot of times it is more convenient to conceptualize the same problem through the conservation of energy. In the simple pendulum below, the total energy of the pendulum remains constant. When the pendulum bob is at it's maximum height (position A or E), all the energy is potential energy, due to gravity. This potential equals the mass of the bob, times acceleration due to gravity, times height (E=mghmax).

Motion of a simple pendulum

As the pendulum swings downward, some of this potential energy is converted downward, its height reduces and some of the potential energy is converted to kinetic energy as the bob gains speed. At it's lowest point (C), the bob has the largest speed - all the potential energy is converted to kinetic energy.

Conservation of Energy

# Temperature

As mentioned earlier in this chapter, temperature is related to motion of molecules. In fact, temperature sets the energy of molecules. Molecules have whats known as 'thermal energy' which defines how fast or slow they move. The thermal energy of molecules is given as below:

Remember the units of Energy is Joules (J for short). T is the temperature in Kelvin and kB is the Boltzmann Constant, that has a value 1.38 × 10-23 J/K. This value might seem very small, which it is at the large scale. This is why you dont see people or cows shaking from thermal energy. However it turns out to be very important at the microscopic scale. In fact if you look under the microscope you will often see stuff jiggling around. This thermal energy is very important in our bodily functions. Multiple processes such as intracellular transport, protein formation, even flexing muscles deal with thermal motion and use it to their advantage.

# Ideal Gas Law

The ideal gas law relates pressure, volume and temperature as below.

Apart from the equation, I like to have a visual picture of any new concept I learn. I've seen this is also very useful for students. Pressure and temperature are proportional to each other. Higher temperature, higher the pressure and vice versa. You can see this from the two visualizations below. Clearly at higher temperature, molecules move much faster. This leads to a higher pressure.

Molecules in gases
Molecules in gases at high temperature
Can you use this information to explain why tire pressure commonly goes low in the winter? The equations below shows the relation between pressure and temperature if volume is kept constant, like in a car tire with fixed volume.

Pressure and volume on the other hand are inversely related. Larger the volume, lower the pressure. Imagine molecules inside a tire bouncing off the internal edges. As temperature increases, molecules hit the surface faster, and at any given moment there are more molecules hitting the surface than at a lower temperature, leading to higher pressure. The converse is true at lower temperature when molecules move slower. But imagine now the volume of the tire is slowly increased. In this scenario, molecules have to travel longer distances to hit the edge, and at any time there are less molecules hitting the surface, leading to lower pressure and the inverse relation between pressure and volume. See the visualization below:

Molecules in gases
Molecules in a large volume
Finally if the number of molecules in the tire increases, say you are pumping more air. This of course increases the pressure and matches your gauge in your tire. See the picture below:
Molecules in gases
Low number of molecules
Pressure is proportional to the number of molecules.

# Example Problems

# Slides

# Lab Activities