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## Physics library

### Course: Physics library > Unit 10

Lesson 1: Temperature, kinetic theory, and the ideal gas law- Thermodynamics part 1: Molecular theory of gases
- Thermodynamics part 2: Ideal gas law
- Thermodynamics part 3: Kelvin scale and Ideal gas law example
- Thermodynamics part 4: Moles and the ideal gas law
- Thermodynamics part 5: Molar ideal gas law problem
- What is the ideal gas law?
- The Maxwell–Boltzmann distribution
- What is the Maxwell-Boltzmann distribution?

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# What is the ideal gas law?

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:**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.**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. minus, 200, start text, space, C, end text) 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.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.

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, equals, 8, point, 31, start fraction, J, divided by, K, dot, m, o, l, end fraction then you must plug in the pressure P in units of start text, p, a, s, c, a, l, s, space, end text, P, a, volume V in units of m, cubed, and temperature T in units of start text, k, e, l, v, i, n, space, end text, K.

If you use the gas constant R, equals, 0, point, 082, start fraction, L, dot, a, t, m, divided by, K, dot, m, o, l, end fraction then you must plug in the pressure P in units of start text, a, t, m, o, s, p, h, e, r, e, s, space, end text, a, t, m, volume V in units of start text, l, i, t, e, r, s, space, end text, L, and temperature T in units of start text, k, e, l, v, i, n, space, end text, K.

This information is summarized for convenience in the chart below.

R, equals, 8, point, 31, start fraction, J, divided by, K, dot, m, o, l, end fraction | R, equals, 0, point, 082, start fraction, L, dot, a, t, m, divided by, K, dot, m, o, l, end fraction | |
---|---|---|

Pressure in start text, p, a, s, c, a, l, s, space, end text, P, a | Pressure in start text, a, t, m, o, s, p, h, e, r, e, s, space, end text, a, t, m | |

Volume in m, cubed | volume in start text, l, i, t, e, r, s, space, end text, L | |

Temperature in start text, k, e, l, v, i, n, space, end text, K | Temperature in start text, k, e, l, v, i, n, space, end text, K |

## What is the molecular form of the ideal gas law?

If we want to use N, start text, space, n, u, m, b, e, r, space, o, f, space, m, o, l, e, c, u, l, e, s, end text instead of n, start text, space, m, o, l, e, s, end text, we can write the ideal gas law as,

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 k, start subscript, B, end subscript is Boltzmann's constant,

When using this form of the ideal gas law with Boltzmann's constant, we have to plug in pressure P in units of start text, p, a, s, c, a, l, s, space, P, a, end text, volume V in start text, m, end text, cubed, and temperature T in start text, k, e, l, v, i, n, space, K, end text. This information is summarized for convenience in the chart below.

k, start subscript, B, end subscript, equals, 1, point, 38, times, 10, start superscript, minus, 23, end superscript, start fraction, J, divided by, K, end fraction | |
---|---|

Pressure in start text, p, a, s, c, a, l, s, space, end text, P, a | |

Volume in m, cubed | |

Temperature in start text, k, e, l, v, i, n, space, end text, K |

## 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 n, R and N, k, start subscript, B, end subscript 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,

This shows that, as long as the number of moles (i.e. molecules) of a gas remains the same, the quantity start fraction, P, V, divided by, T, end fraction 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 P, start subscript, 1, end subscript, volume V, start subscript, 1, end subscript, and temperature T, start subscript, 1, end subscript) and is altered to a state 2 (with P, start subscript, 2, end subscript, volume V, start subscript, 2, end subscript, and temperature T, start subscript, 2, end subscript), then regardless of the details of the process we know the following relationship holds.

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 start text, m, end text, cubed for V, start subscript, 1, end subscript, we'll have to use start text, m, end text, cubed for V, start subscript, 2, end subscript). [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, point, 54, start text, space, a, t, m, end text and the ball has a radius of 0, point, 119, start text, space, m, end text. Assume the temperature of the air inside the basketball is 25, start superscript, o, end superscript, start text, space, C, end text (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.

Given this choice of gas constant, we need to make sure we use the correct units for pressure (start text, p, a, s, c, a, l, s, end text), volume (start text, m, end text, cubed), and temperature (start text, k, e, l, v, i, n, end text).

We can convert the pressure as follows,

1, point, 54, start text, space, a, t, m, end text, times, left parenthesis, start fraction, 1, point, 013, times, 10, start superscript, 5, end superscript, start text, space, P, a, end text, divided by, 1, start text, space, a, t, m, end text, end fraction, right parenthesis, equals, 156, comma, 000, start text, space, P, a, end text.

1, point, 54, start text, space, a, t, m, end text, times, left parenthesis, start fraction, 1, point, 013, times, 10, start superscript, 5, end superscript, start text, space, P, a, end text, divided by, 1, start text, space, a, t, m, end text, end fraction, right parenthesis, equals, 156, comma, 000, start text, space, P, a, end text.

And we can use the formula for the volume of a sphere start fraction, 4, divided by, 3, end fraction, pi, r, cubed to find the volume of the gas in the basketball.

V, equals, start fraction, 4, divided by, 3, end fraction, pi, r, cubed, equals, start fraction, 4, divided by, 3, end fraction, pi, left parenthesis, 0, point, 119, start text, space, m, end text, right parenthesis, cubed, equals, 0, point, 00706, start text, space, m, end text, cubed

V, equals, start fraction, 4, divided by, 3, end fraction, pi, r, cubed, equals, start fraction, 4, divided by, 3, end fraction, pi, left parenthesis, 0, point, 119, start text, space, m, end text, right parenthesis, cubed, equals, 0, point, 00706, start text, space, m, end text, cubed

The temperature 25, start superscript, o, end superscript, start text, space, C, end text can be converted with,

T, start subscript, K, end subscript, equals, T, start subscript, C, end subscript, plus, 273, start text, space, K, end text. T, equals, 25, start superscript, o, end superscript, start text, space, C, end text, plus, 273, start text, space, K, end text, equals, 298, start text, space, K, end text.

T, start subscript, K, end subscript, equals, T, start subscript, C, end subscript, plus, 273, start text, space, K, end text. T, equals, 25, start superscript, o, end superscript, start text, space, C, end text, plus, 273, start text, space, K, end text, equals, 298, start text, space, K, end text.

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

Now to determine the number of air molecules N in the basketball we can convert start text, m, o, l, e, s, end text into start text, m, o, l, e, c, u, l, e, s, end text.

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, equals, 293, start text, space, K, end text and atmospheric pressure. The canister is then placed in an ice bath and allowed to cool to a temperature of T, equals, 255, start text, space, K, end text.

**Determine the pressure of the gas after reaching a temperature of 255, start text, space, K, end text, point**

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.

Notice that we plugged in the pressure in terms of start text, a, t, m, o, s, p, h, e, r, e, s, end text and ended up with our pressure in terms of start text, a, t, m, o, s, p, h, e, r, e, s, end text. If we wanted our answer in terms of start text, p, a, s, c, a, l, s, end text we could have plugged in our pressure in terms of start text, p, a, s, c, a, l, s, end text, or we can simply convert our answer to start text, p, a, s, c, a, l, s, end text as follows,

## Want to join the conversation?

- Where do
*R, Na(Avogadro's Number) and k(Boltzmann's constant)*come from and why? Is there an explanation for how they have been calculated? Thanks in advance. I wouldn't mind if the answer involved calculus.(5 votes)- No calculus needed :-) Like most any constants, they are simply needed if there is always that same factor missing in an equation.

For example, in statistical mechanics you have a formula that is: S=k*ln(W). If you know S and W for at least two cases, then you might realize that,*for both cases,*S = ln(W) only if you multiply the right side by k constant(6 votes)

- When converting, why should we use Kelvin?(2 votes)
- One of the most important formulas in thermodynamics is P1 * V1 / T1= P2 * V2 / T2. However, if we used Celsius or Fahrenheit, what if, for example, the temperature was 0 degrees Celsius? Since you can't divide by 0, the formula would not work.

The Kelvin scale is made with 0 being equal to absolute zero, the coldest possible temperature, where the molecules stop moving completely. Therefore, you will never get a zero or negative temperature in your formula if you use Kelvin.

Kelvin is also the widely accepted temperature scale. If, for example, some people used Celsius and some people used Kelvin, we would all get different answers, so everyone uses Kelvin.(8 votes)

- In the section "What is the molar form of the ideal gas law?" and the first example, shouldn't the atm version of the ideal gas constant be 0.082 L*atm/mol*K instead of 0.082 L*atm/K? Or is there some reason the number of moles isn't included?(4 votes)
- You are right, the R actually does have the "mol" units, and it should read, as you correctly mentioned, L*atm/mol*K.

Nevertheless, the reason why this was probably excluded here is because the units of n are mol, and then if you combine n and R, the mol units will cancel.(4 votes)

- How do I know when a gas behaves like an ideal gas?(4 votes)
- most real gases do as long as the temperature is not too low and the pressure is not too high(2 votes)

- I know that Charles Law need constant moles and constant temperature; Boyles' law needs constant moles and constant temperature; so what does Avogadro's Law and Gay-Lussac's law need?(4 votes)
- Gay-Lussac's law has a constant volume. 'For a given volume of a gas, as the temperature increases, the pressure of the gas is directly proportional'. Volume is not a variable in his formula.

Avogadro's law

1 mol = 6.02 x 10^23

1 mol = 22.4 L

1 mol = molar mass in grams

Avogadro's law is mostly used for converting from one unit to another so the constant will depend on what you are converting.

What it does require is that you use the correct unit of measurement. For example, you have to use liters, you can't use milliliters.(1 vote)

- how does the K.E transfer between two molecules (elastic collision) and no loss of energy ?(3 votes)
- That is the definition of an elastic collision. Remember: this is an ideal scenario. Nothing like this ever happens in real life. So when we talk about elastic collisions, we are taking the kinetic energy as conserved and then finding appropriate values of velocities that would allow the kinetic energy to be conserved
**and**simultaneously obey the law of conservation of momentum.

I suggest watching these videos if you're a little rusty on collision theory.

https://www.khanacademy.org/science/physics/linear-momentum/elastic-and-inelastic-collisions/v/elastic-and-inelastic-collisions(1 vote)

- In the "Units to use for PV=nRT" section, It says 1 liter=0.001 m^3=1000 cm^3.

This doesn't make sense to me. Isn't 1000 cm^3 = 10m^3 since c is a SI prefix for 10^-2?

10m^3 isn't equal to 0.001m^3. What am I missing? I'm sorry if this is a silly question.(1 vote)- Your math is a little bit wrong. Check it:

1 cm = (10^-2) m

(1 cm)^3 = (10^-2 m )^3

1 cm^3 = (10^-6) m^3

(1 cm^3)*1000 = (10^-6) m^3 *1000

1000 cm^3 = (10^-6)*(10^3) m^3

1000 cm^3 = 10^-3 m^3 = 0.001 m^3(5 votes)

- Where do we get the gas constant ,R, from?

Thanks(0 votes)- Choose any gas, assuming its ideal. For example, 1 mole of Ar = 39.948 = 22.4 L at standard pressure ( 1 atm)

Just solve for R using the same formula, PV=nRT or in this case, R = PV/nT

Subsitue values into the equations :- R = (1atm) (22.4) / (1mole) (273K)

Solve: R = 0.0821 atm L / mol K

There are also alternate values for different units.

If R is needed in units of pressure (kPa) = 8.314 kPa L / mol K

If R is needed in units of pressure (mm Hg) = 62.396 mm Hg L / mol K

I would suggest always using the value of 0.0821 atm L / mol K for R unless stated otherwise.

Hope this helped!(5 votes)

- In all these video on Thermodynamics from part 1 to part 5

Pressure mean here absolute pressure or gauge pressure?

What is NTP(Normal Temperature and Pressure)?(2 votes) - What factor is found in the ideal gas law which is not in the previous laws?(1 vote)
- Nothing extra. The ideal gas law is the integration of Boyle's, Charles' and Avogadro's laws into a single equation.(2 votes)