What is the ideal gas law? (article) | Gases | Khan Academy (2024)

Learn how pressure, volume, temperature, and the amount of a gas are related to each other.

What is an ideal gas?

Gases are complicated. They're full of billions and billions of energetic gas molecules that can collide and possibly interact with each other. Since it's hard to exactly describe a real gas, people created the concept of an Ideal gas as an approximation that helps us model and predict the behavior of real gases. The term ideal gas refers to a hypothetical gas composed of molecules which follow a few rules:

  1. Ideal gas molecules do not attract or repel each other. The only interaction between ideal gas molecules would be an elastic collision upon impact with each other or an elastic collision with the walls of the container.

    The phrase elastic collision refers to a collision wherein no kinetic energy is converted to other forms of energy during the collision. In other words, kinetic energy can be exchanged between the colliding objects (e.g. molecules), but the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

    A car crash where kinetic energy gets converted into heat energy and sound energy yielding two crumpled bumpers would not be elastic.

  2. Ideal gas molecules themselves take up no volume. The gas takes up volume since the molecules expand into a large region of space, but the Ideal gas molecules are approximated as point particles that have no volume in and of themselves.

If this sounds too ideal to be true, you're right. There are no gases that are exactly ideal, but there are plenty of gases that are close enough that the concept of an ideal gas is an extremely useful approximation for many situations. In fact, for temperatures near room temperature and pressures near atmospheric pressure, many of the gases we care about are very nearly ideal.

If the pressure of the gas is too large (e.g. hundreds of times larger than atmospheric pressure), or the temperature is too low (e.g. 200C) there can be significant deviations from the ideal gas law. For more on non-ideal gases read this article.

What is the molar form of the ideal gas law?

The pressure, P, volume V, and temperature T of an ideal gas are related by a simple formula called the ideal gas law. The simplicity of this relationship is a big reason why we typically treat gases as ideal, unless there is a good reason to do otherwise.

PV=nRT

Where P is the pressure of the gas, V is the volume taken up by the gas, T is the temperature of the gas, R is the gas constant, and n is the number of moles of the gas.

Molesn are a way to describe how many molecules are in a gas.

1mole is equal to 6.02×1023molecules. This number is called Avogadro's constant NA and it's a way to convert from moles to molecules or vice versa.

NA=6.02×1023moleculesmole

For a typical room there is likely to be at least 1000moles of gas molecules. That's an almost unthinkably high number of molecules since,

1000×6.02×1023molecules=6.02×1027=six billion billion billion gas molecules

This is far larger than the estimated number of stars in the Milky Way galaxy, and is even larger than most estimates of the number of stars in the observable universe.

Perhaps the most confusing thing about using the ideal gas law is making sure we use the right units when plugging in numbers. If you use the gas constant R=8.31JKmol then you must plug in the pressure P in units of pascalsPa, volume V in units of m3, and temperature T in units of kelvinK.

If you use the gas constant R=0.082LatmKmol then you must plug in the pressure P in units of atmospheresatm, volume V in units of litersL, and temperature T in units of kelvinK.

This information is summarized for convenience in the chart below.

Units to use for PV=nRT
R=8.31JKmolR=0.082LatmKmol
Pressure in pascalsPaPressure in atmospheresatm
Volume in m3volume in litersL
Temperature in kelvinKTemperature in kelvinK

Here is some useful information relating the different types of units.

1atmosphere=1.013×105Pa=14.7pounds per square inch

1liter=0.001m3=1000cm3

0oC=273K=32oF

To convert celsius into kelvin we can use the formula TK=TC+273K.

Also, the term STP refers to "standard temperature and pressure" which are defined to be T=273K=0ocelsius and P=1.013×105Pa=1atm respectively.

What is the molecular form of the ideal gas law?

If we want to use Nnumber of molecules instead of nmoles, we can write the ideal gas law as,

PV=NkBT

Where P is the pressure of the gas, V is the volume taken up by the gas, T is the temperature of the gas, N is the number of molecules in the gas, and kB is Boltzmann's constant,

kB=1.38×1023JK

When using this form of the ideal gas law with Boltzmann's constant, we have to plug in pressure P in units of pascals Pa, volume V in m3, and temperature T in kelvin K. This information is summarized for convenience in the chart below.

Units to use for PV=NkBT
kB=1.38×1023JK
Pressure in pascalsPa
Volume in m3
Temperature in kelvinK

What is the proportional form of the ideal gas law?

There's another really useful way to write the ideal gas law. If the number of moles n (i.e. molecules N) of the gas doesn't change, then the quantity nR and NkB are constant for a gas. This happens frequently since the gas under consideration is often in a sealed container. So, if we move the pressure, volume and temperature onto the same side of the ideal gas law we get,

nR=NkB=PVT=constant

This shows that, as long as the number of moles (i.e. molecules) of a gas remains the same, the quantity PVT is constant for a gas regardless of the process through which the gas is taken. In other words, if a gas starts in state 1 (with some value of pressure P1, volume V1, and temperature T1) and is altered to a state 2 (with P2, volume V2, and temperature T2), then regardless of the details of the process we know the following relationship holds.

P1V1T1=P2V2T2

This formula is particularly useful when describing an ideal gas that changes from one state to another. Since this formula does not use any gas constants, we can use whichever units we want, but we must be consistent between the two sides (e.g. if we use m3 for V1, we'll have to use m3 for V2). [Temperature, however, must be in Kelvins]

What do solved examples involving the ideal gas law look like?

Example 1: How many moles in an NBA basketball?

The air in a regulation NBA basketball has a pressure of 1.54atm and the ball has a radius of 0.119m. Assume the temperature of the air inside the basketball is 25oC (i.e. near room temperature).

a. Determine the number of moles of air inside an NBA basketball.

b. Determine the number of molecules of air inside an NBA basketball.

We'll solve by using the ideal gas law. To solve for the number of moles we'll use the molar form of the ideal gas law.

PV=nRT(use the molar form of the ideal gas law)

n=PVRT(solve for the number of moles)

n=PV(8.31JKmol)T(decide which gas constant we want to use)

Given this choice of gas constant, we need to make sure we use the correct units for pressure (pascals), volume (m3), and temperature (kelvin).

Yes, we could have used the gas constant R=0.082LatmKmol. We would have just had to be careful to plug in pressure in terms of atmospheres and volume in terms of liters.

We can convert the pressure as follows,
1.54atm×(1.013×105Pa1atm)=156,000Pa.

And we can use the formula for the volume of a sphere 43πr3 to find the volume of the gas in the basketball.
V=43πr3=43π(0.119m)3=0.00706m3

The temperature 25oC can be converted with,
TK=TC+273K. T=25oC+273K=298K.

Now we can plug these variables into our solved version of the molar ideal gas law to get,

n=(156,000Pa)(0.00706m3)(8.31JKmol)(298K)(plug in correct units for this gas constant)

n=0.445moles

Now to determine the number of air molecules N in the basketball we can convert moles into molecules.

N=0.445moles×(6.02×1023molecules1mole)=2.68×1023molecules

Alternatively, we could have solved this problems by using the molecular version of the ideal gas law with Boltzmann's constant to find the number of molecules first, and then converted to find the number of moles.

Example 2: Gas takes an ice bath

A gas in a sealed rigid canister starts at room temperature T=293K and atmospheric pressure. The canister is then placed in an ice bath and allowed to cool to a temperature of T=255K.

Determine the pressure of the gas after reaching a temperature of 255K.

Since we know the temperature and pressure at one point, and are trying to relate it to the pressure at another point we'll use the proportional version of the ideal gas law. We can do this since the number of molecules in the sealed container is constant.

P1V1T1=P2V2T2(start with the proportional version of the ideal gas law)

P1VT1=P2VT2(volume is the same before and after since the canister is rigid)

P1T1=P2T2(divide both sides byV)

P2=T2P1T1(solve for the pressureP2)

P2=(255K)1atm293K(plug in values for pressure and temperature)

P2=0.87atm(calculate and celebrate)

Notice that we plugged in the pressure in terms of atmospheres and ended up with our pressure in terms of atmospheres. If we wanted our answer in terms of pascals we could have plugged in our pressure in terms of pascals, or we can simply convert our answer to pascals as follows,

P2=0.87atm×(1.013×105Pa1atm)=88,200Pa(convert from atmospheres to pascals)

What is the ideal gas law? (article) | Gases | Khan Academy (2024)
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