### Video Transcript

This lesson is about standard molar
gas volumes or, in simpler terms, under certain conditions, how much space does a
gas take up? In this lesson, we will investigate
the relationship between the volume, temperature, pressure, and amount in moles of a
gas. We will learn how to calculate the
molar volume of a gas as well as what constitutes standard conditions. We will also use molar volume and
the ideal gas law to calculate the characteristics of a given gas.

Letโs imagine for a moment a big
pink balloon. As a thought experiment, what are
the ways that we could change the volume of the balloon? Well, the simplest way to change
the volume is to change the amount in moles of gas in the balloon. Blow the balloon up and the volume
will increase, or let some air out of the balloon and the volume will decrease. We can also change the volume by
changing the temperature. If we put the balloon in the
fridge, it will shrivel and shrink. Leave it in the sun and the balloon
will swell in size.

Lastly, we can also change the air
pressure of the room, which will cause the air pressure in the balloon to change and
cause the volume to change. We could increase the pressure and
reduce the volume by pulling the balloon under water. And we could decrease the pressure
and increase the volume by climbing into the low-pressure air atop a mountain. In this video, we will see how
these four characteristics of a gas, amount in moles, temperature, pressure, and
volume, are related. We will also use some key equations
to calculate their values. First, some key concepts.

The molar volume of a gas is the
volume occupied by every mole of the gas. Itโs given by the formula ๐ m
equals ๐ divided by ๐, where ๐ m is the molar volume, ๐ is the volume occupied
by the gas as a whole, and ๐ is the amount in moles of that gas. For example, if one mole of a gas
has a volume of 24 liters, the volume per mole is 24 liters divided by one mole
equals 24 liters per mole. If we have a smaller 2.4-liter
container of a gas with 0.1 moles of the gas in it, the molar volume of that gas is
2.4 liters divided by 0.1 moles equals also 24 liters per mole. This unit of liters per mole looks
similar to some other units that we might recognize from other topics in
chemistry. The molar volume is the volume per
mole of a gas.

On the other hand, molar mass is
the number of grams in a mole of a substance. We can apply molar mass to any
state of matter, including gases. Molar concentration, often
expressed as moles per liter or molarity, is a measurement of the amount of moles of
a solute dissolved in a unit of volume of a solvent. We most frequently see grams for
mass and liters for volume, but we may see other units as well. The molar volume gives us the ratio
of liters to moles. If we know the molar volume and
either the amount in moles or the volume, we can solve for the missing value by
using a rearranged version of the original formula.

For example, if we have another
container of 12 liters of the gas with a molar volume of 24 liters per mole, we can
use the formula ๐ equals ๐ divided by ๐ m to calculate that 12 liters divided by
24 liters per mole equals 0.5 moles. This answer also makes sense
intuitively. If every 24 liters of gas is one
mole, then half that amount, 12 liters, should be half a mole of the gas.

As another example, if we put
exactly 0.25 moles of a gas with a molar volume of 24 liters per mole into a
balloon, that balloon would have a volume of 24 liters per mole times 0.25 moles
equals six liters. Note that since the volume of a gas
will change depending on the temperature and pressure, the molar volume will change
as well. That means that thereโs no one
consistent molar volume for a specific gas like oxygen. It will depend on the temperature
and the pressure of the room the oxygen is in. In fact, the type of gas generally
does not affect the molar volume.

As weโve discussed a bit already,
the different characteristics of gases are dependent on one another. Over the years, chemists have
noticed different simple relationships between the characteristics of gases and
developed laws about them. In 1662, Robert Boyle discovered
Boyleโs law, which states that all other things equal, the pressure and volume of a
gas are inversely proportional. This means that if the pressure or
volume goes up, the other will go down. For example, a balloon will shrink
or decrease in volume under increased pressure, like when itโs pulled
underwater.

In 1787, Jacques Charles developed
Charlesโs law, which states that all other things equal, the temperature of a gas in
kelvin and its volume are directly proportional. This means that temperature and
volume will increase together and they will decrease together. For example, a balloon will swell
and increase in volume if it is heated in the sun and increase in temperature. In 1808, Joseph Gay-Lussac
discovered Gay-Lussacโs law, which states that all other things equal, the
temperature in kelvin and the pressure of the gas will be directly proportional. As an example, in a container where
the volume cannot change like a propane tank, if the temperature of the gas is
increased, the pressure will increase as well because the heated particles will
collide with the container with more energy.

And in 1811, Amedeo Avogadro
proposed Avogadroโs law, which states that two different gases with the same volume,
temperature, and pressure will have the same number of particles. Avogadroโs law allows us to draw
conclusions about the amount in moles from the other characteristics. Amazingly, all of these individual
relationships can be combined together in a single equation that incorporates
pressure, volume, temperature, and amount in moles. That equation is called the ideal
gas law, and it reads as such: ๐๐ equals ๐๐
๐, where ๐ is the pressure, ๐ is
the volume, ๐ is the amount in moles, ๐
is the gas constant, and ๐ is the
temperature in kelvin.

The trickiest part of the equation,
๐
, is a constant that acts as a unit conversion factor, often in liter bars per
mole kelvin. Why do we need to include a
constant with this bizarre unit? Well, from a purely unit-based
perspective, in order for this equation to be truly equal, the left and right sides
must have the same unit. When we look at how the units
cancel out, we can see that the conversion factor makes that equivalence happen. As a deeper explanation, ๐
is the
conversion factor that mathematically links all four of the individual
characteristics weโre examining. ๐
tells us the precise,
quantifiable, and natural relationship between the pressure and volume and the
amount and temperature of an ideal gas so we can actually convert from one to
another rather than just using the individual relationships from the gas laws.

Since ๐
is a constant, the value
will not change. But it might appear as a different
number if we change its units. Just as 100 centimeters and one
meter are the same length represented by different units, 0.08315 liter bars per
mole kelvin and 0.08206 liter atmospheres per mole kelvin, the two most commonly
used numbers for ๐
, are the same unit conversion factor with different units. The equation as a whole reveals the
relationship between the different characteristics within. Mainly, there is a direct
relationship between variables on opposite sides of the equation, such as pressure
and temperature, and an indirect relationship between variables on the same side of
the equation, such as pressure and volume.

We can think back to our balloon
example to put this equation into context. Letโs imagine that we heat up the
balloon, increasing its temperature. Well, as a result, in order for the
equation to remain equal, the left side of the equation must increase as well,
either by increasing in pressure, volume, or both. Letโs also imagine a situation
where we squeeze in on the balloon. In this situation, weโre decreasing
the volume available. One way for the gas to compensate
for this would be an increase in pressure. The decrease on the left side of
the equation can be balanced out by an increase on the same side.

Again, there is a direct
relationship between variables on opposite sides of the equation and an indirect
relationship between variables on the same side of the equation. Note that none of these variables
represent anything about the type of gas involved. The volume of two different ideal
gases with the same temperature, pressure, and amount in moles will be equal. This equation is called the ideal
gas law. What makes a gas ideal? An ideal gas is a simplified
theoretical version of a gas that lets us more easily carry out calculations. Gas particles in real life attract
one another and take up a minuscule amount of space, whereas the simplified ideal
gas particles are assumed to not interact with one another at all and take up no
space.

While these assumptions are
technically inaccurate, these traits have such a small effect on the gas that we can
ignore them at low temperatures and pressures to simplify the math involved. The simple takeaway, an ideal gas
is a simplified gas. One question that may arise when
looking at gases is if traits including molar volume depend on the temperature and
pressure which vary from place to place, wouldnโt it be difficult to make
conclusions about gases? How do we overcome this
variance? Well, thatโs why scientists
developed some standards for temperature and pressure. These standards set baseline values
for temperature and pressure so that we can more easily make calculations and more
easily compare different gases.

We can easily reference a standard
by name and then immediately know the resulting pressure, temperature, and molar
volume of that standard. Letโs take a look at some of the
values associated with these named standards. STP, or standard temperature and
pressure, refers to a pressure of one bar, a temperature of zero degrees Celsius,
and a molar volume of 22.7 liters per mole. However, in other sources, you may
also see STP refer to a pressure of one atmosphere. This changes the molar volume to
22.4 liters per mole. Both of these sets of numbers are
correct. Youโll simply see different
conventions used in different places.

SATP, or standard ambient
temperature and pressure, refers to a pressure of one bar, a temperature of 25
degrees C, and a molar volume of 24.8 liters per mole. RTP, or room temperature and
pressure, refers to a pressure of one atmosphere, a temperature of 20 degrees C, and
a molar volume of 24.0 liters per mole. An NTP, or normal temperature and
pressure, refers to the same values as RTP. When we answer a question about
gases, we will likely see one of these named standards in the wording of the
question telling us the temperature, pressure, and molar volume we can use in any
calculations. STP and RTP are the most commonly
used standards. Note that as the temperature of the
standard increases, so does the molar volume. This makes sense, as a hotter gas
will take up more space.

The two units of pressure that we
see here, bar and atmosphere, are very similar in value. One atmosphere equals 1.013
bars. One atmosphere, which isnโt much
different from one bar, is the approximate air pressure at sea level on Earth. Itโs simply convention whether we
choose to use atmosphere or bar, so pay attention to the wording of the question to
know which one to use. And itโs worth noting again that
different textbooks might use slightly different definitions for each standard. Now that weโve learned about molar
volumes, the ideal gas law, and standards, letโs do some practice that ties them all
together.

What is the molar volume of a gas
at standard temperature and pressure, to two significant figures?

This value is commonly listed as a
reference value. In this problem, weโre simply
deriving that value so we know where it comes from. If you have the molar volume of a
gas at standard temperature and pressure memorized, you can use that reference value
without doing the math shown here. But for this problem, we will carry
out the calculations. As a reminder, molar volume is the
number of liters taken up by a mole of the gas. Standard temperature and pressure
refers to a temperature of zero degrees C and a pressure of one bar. Since we want our temperature to be
in kelvin instead of degrees C, we simply add 273 to our value in degrees C to find
that temperature in kelvin.

The formula for molar volume is ๐
m equals ๐ divided by ๐, where ๐ m equals the molar volume, ๐ equals the volume,
and ๐ equals the amount in moles. However, we donโt know the volume
or the amount in moles, so we canโt carry out the calculation directly. However, we do know the pressure,
the temperature, and the value of the gas constant. If we look at the ideal gas law, we
can put ๐ over ๐ in terms of numbers that we already know to find a value for ๐
over ๐. If we use algebra and divide both
sides of the equation by ๐ times ๐, we end up with the equation ๐ over ๐ equals
๐
๐ over ๐. Weโve grouped all of the variables
that we donโt know the value of on the left side of the equation and all the
variables that we do know the value of on the right side of the equation.

We know that the molar volume
equals ๐ over ๐. We donโt know directly the value of
๐ over ๐, but we do know that it equals ๐
๐ over ๐, a value that we can
calculate. We wanna use the value of ๐
that
matches the units that weโre using, namely, liters and bars. So we wanna use the value ๐
equals
0.8315 liter bars per mole kelvin. If we plug our known values back
into the equation, we get 0.8315 liter bar per mole kelvin times 273 kelvin divided
by one bar. If we carry out the arithmetic, we
arrive at our final answer.

Note that if we had used the
alternate value of one atmosphere instead of one bar for the pressure at STP, we
would use a value of ๐
with different units, which would change our final
answer. In this case, weโve used a value of
one bar for the pressure and calculated the molar volume of a gas at standard
temperature and pressure to be 22.7 liters per mole.

Letโs review with some key
points. Molar volume is the volume per one
mole of a gas, often expressed in the unit liters per mole. We can calculate the molar volume
by dividing the volume by the number of moles. The ideal gas law ๐๐ equals
๐๐
๐ relates the pressure, volume, number of moles, and temperature of the
gas. The two most common standards are
STP, referring to a pressure of one bar, a temperature of zero degrees Celsius, and
a molar volume of 22.7 liters per mole, and RTP, referring to a pressure of one
atmosphere, a temperature of 20 degrees Celsius, and a molar volume of 24.0 liters
per mole.