iDEAL gAS lAW

Ideal Gas Law

• An Ideal Gas (perfect gas)is one which obeys Boyle's Law and Charles' Law exactly.
• An Ideal Gas obeys the Ideal Gas Law (General gas equation):

PV = nRT

where
    P=pressure
    V=volume
    n=moles of gas
    T=temperature
    R = gas constant (dependent on the units of pressure, temperature and volume)
        R = 8.314 J K^-1 mol^-1 if
            Pressure is in kilopascals(kPa)
            Volume is in litres(L)
            Temperature is in kelvin(K)
        R = 0.0821 L atm K^-1 mol^-1 if
            Pressure is in atmospheres(atm)
            Volume is in litres(L)
            Temperature is in kelvin(K)

• An Ideal Gas is modelled on the Kinetic Theory of Gases which has 4 basic postulates:

1. Gases consist of small particles (molecules) which are in continuous random motion
2. The volume of the molecules present is negligible compared to the total volume occupied by the gas
3. Intermolecular forces are negligible
4. Pressure is due to the gas molecules colliding with the walls of the container

• Real Gases deviate from Ideal Gas Behaviour because:

1. at low temperatures the gas molecules have less kinetic energy (move around less) so they do attract each other
2. at high pressures the gas molecules are forced closer together so that the volume of the gas molecules becomes significant compared to the volume the gas occupies

• Under ordinary conditions, deviations from Ideal Gas behaviour are so slight that they can be neglected
• A gas which deviates from Ideal Gas behaviour is called a non-ideal gas.

Ideal Gas Law Calculations

Calculating Volume of Ideal Gas: V = (nRT) ÷ P

What volume is needed to store 0.050 moles of helium gas at 202.6kPa and 400K?

PV = nRT

P = 202.6 kPa
n = 0.050 mol
T = 400K
V = ? L
R = 8.314 J K^-1 mol^-1

202.6V = 0.050 x 8.314 x 400
202.6 V = 166.28
V = 166.28 ÷ 202.6
V = 0.821 L (821mL)

Calculating Pressure of Ideal Gas: P = (nRT) ÷ V

What pressure will be exerted by 20.16g hydrogen gas in a 7.5L cylinder at 20^oC?

PV = nRT

P = ? kPa
V = 7.5L
n = mass ÷ MM
  mass = 20.16g
  MM(H₂) = 2 x 1.008 = 2.016g/mol
n = 20.16 ÷ 2.016 = 10mol
T = 20^o = 20 + 273 = 293K
R = 8.314 J K^-1 mol^-1

P x 7.5 = 10 x 8.314 x 293
P x 7.5 = 24360.02
P = 24360.02 ÷ 7.5 = 3248kPa

Calculating moles of gas: n = (PV) ÷ (RT)

A 50L cylinder is filled with argon gas to a pressure of 10130.0kPa at 30^oC. How many moles of argon gas are in the cylinder?

PV = nRT

P = 10130.0kPa
V = 50L
n = ? mol
R = 8.314 J K^-1 mol^-1
T = 30^oC = 30 + 273 = 303K

10130.0 x 50 = n x 8.314 x 303
506500 = n x 2519.142
n = 506500 ÷ 2519.142 = 201.1mol

Calculating gas temperature: T = (PV) ÷ (nR)

To what temperature does a 250mL cylinder containing 0.40g helium gas need to be cooled in order for the pressure to be 253.25kPa?

PV = nRT

P = 253.25kPa
V = 250mL = 250 ÷ 1000 = 0.250L
n = mass ÷ MM
  mass = 0.40g
  MM(He) = 4.003g/mol
n = 0.40 ÷ 4.003 = 0.10mol
R = 8.314 J K mol^-1
T = ? K

253.25 x 0.250 = 0.10 x 8.314 x T
63.3125 = 0.8314 x T
T = 63.3125 ÷ 0.8314 = 76.15K

Charles' Law:

Assuming that pressure remains constant, the volume and absolute temperature of a certain quantity of a gas are directly proportional.

Mathematically, this can be represented as:

Temperature = Constant x Volume
or
Volume = Constant x Temperature
or
Volume/Temperature = Constant

Substituting in variables, the formula is:

V/T=K

Because the formula is equal to a constant, it is possible to solve for a change in volume or temperature using a proportion:

V/T = V₁/T₁

Explanation and Discussion:

Charles' Law describes the direct relationship of temperature and volume of a gas. Assuming that pressure does not change, a doubling in absolute temperature of a gas causes a doubling of the volume of that gas. A drop of absolute temperature sees a proportional drop in volume. The volume of a gas increases by 1/273 of its volume at 0°C for every degree Celsius that the temperature rises.

To explain why this happens, let's explore temperature and volume in terms of gases. Temperature is an average of molecular motion. This means that, while all of the gas molecules are moving around their container in different directions at different speeds, they will have an average amount of energy that is the temperature of the gas. The volume of the gas is the size of its container because the molecules will move in a straight line until they impact something (another molecule or the container). However, to move as they do, the molecules require kinetic energy, which is measured by temperature.

So, the volume and temperature are very closely related. If the temperature was not sufficient, the molecules would not be able to overcome the weak forces of attraction among them and would not be able to fill the container.

Charles' Law must be used with the Kelvin temperature scale. This scale is an absolute temperature scale. At 0 K, there is no kinetic energy (Absolute Zero). According to Charles' Law, there would also be no volume at that temperature. This condition cannot be fulfilled because all known gases will liquify or solidify before reaching 0 K. The Kelvin temperature scale is Celcius minus 273.15 °. Therefore, zero Kelvin would be -273.15 ° and any Celcius temperature can be converted by to Kelvin by adding 273.15 (273 is often used).

Any unit of volume will work with Charles' Law, but the most common are liters (dm³) and milliliters (cm³).

Calculations with Charles' Law

Let's try a problem with Charles' Law. For example, let's try to solve for an unknown volume of a gas. The unknown volume is at 32°C. At 18°C the gas occupied a volume of 152 mL.

Set-up

First, we must convert degrees Celcius to Kelvins. To do this, we add 273 to the Celcius measure. So:

32°C + 273 = 305 K
18°C + 273 = 291 K

Estimate answer

We know that the temperature and volume are directly related. The temperature only went up a little bit (slightly more than 5%). So, we can expect the volume to increase by about 5%, which would be about 7.5 mL. Now, we can use the formula. (Really, we should use fraction ratios.)

Plug values into formula

Our formula is: V/T = V₁/T₁

In this problem, V = 152 mL, T = 291 K, and T₁ = 305 K. V₁ is unknown. Therefore, we can arrange the formula as:

152 mL/291 K = ? /305 L

Because this is a direct proportion, we can multiple the means and extremes to create an ease to solve equation:

152 mL x 305 K = 291 K x ?

Which can be divided by 291 K to yield:

152 mL x 305 K / 291 K = Volume₁ = 159.3127 mL

However, we only have three degrees of precision in this problem, so our answer is: Volume₁ = 159 mL.

Check

To check our answer, we need to compare it to our earlier estimate. We expected the volume to increase by about 7.5 mL, and it increased by 7 mL (7.3 before round). This answer is acceptable.

Continued Study
For continued study, you can visit our Charles' Law bonus page. You can also test yourself. You can also learn about Jacques Charles.

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Sources:

Brown, Theodore L., H. Eugene LeMay, Jr. and Bruce E. Burston, Chemistry: The Central Science, Englewood Cliffs, NJ: Prentice Hall, Inc., 1994

Dorin, Henry, Peter E. Demmin, and Dorothy L. Gabel. Prentice Hall Chemistry: The Study of Matter, Needham, Massachusetts and Englewood Cliffs, New Jersey: Prentice Hall, Inc., 1989.

Roper, Gerald C., "gas laws" Groliers New Multimedia Encyclopedia, Release 6, 1993

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Choose the best answer.

Top of Form

Question 1: 5.0g of an ideal gas occupies 9.2 L at STP. What volume would it occupy at 120°C?

13 L

3 L

130 L

None of These

Question 2: 5.0g of an ideal gas occupies 9.2 L at STP. What volume would it occupy at 0°C and 93 mm Hg?

7.5 L

10 L

75 L

None of These

Question 3: 5.0g of an ideal gas occupies 9.2 L at STP. What is the molecular mass of the substance?

12 g

13 g

78 g

None of These

Question 4: 5.0g of an ideal gas occupies 9.2 L at STP. What volume would it occupy at 120°C and 92 mm Hg?

19 L

109 L

0.42 L

None of These

Question 5: Calculate the volume occupied by 12.0 g of CO₂ gas at 245 mm Hg and -35 °C.

5 L

10 L

15 L

None of These

Question 6: A sample of an ideal gas occupies 1.4 L at 28 cm Hg and 52 degrees C. Calculate the number of moles in the solution.

19 mol

0.019 mol

1.9 mol

None of these

Question 7: An Ideal gas has a volume of 15.0 L at 15°C and 735 mm Hg pressure. At what temperature would it occupy a volume of 30.0 L at 785 mm Hg?

615 K

512 K

128 K

None of these

Question 8: A sample of ideal gas occupies a volume of 238 mL at STP. To what temperature must the sample be heated if it is to occupy a volume of 185 mL at 2.25 atm?

94.3 K

477 K

477°C

None of these

Question 9: Increasing the pressure on a sample of gas increases its temperature, but its volume remains the same. What law best explains this?

Avogadro's Law

Graham's Law

Ideal Gas Law

Avogadro's Interpretation of Guy-Lassac's observations

Question 10: A few minutes after opening a bottle of perfume, the scent permeates the room. What law relates to this phenomenon?

Boyle's Law

Graham's Law

Charles' Law

None of these

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Boyle's law

Boyle's law shows that, at constant temperature, the product of an ideal gas's pressure and volume is always constant. It was published in 1662. It can be determined experimentally using a pressure gauge and a variable volume container. It can also be found through the use of logic; if a container, with a fixed number of molecules inside, is reduced in volume, more molecules will hit the sides of the container per unit time, causing a greater pressure.

As a mathematical equation, Boyle's law is:

where P is the pressure (Pa), V the volume (m³) of a gas, and k₁ (measured in joules) is the constant from this equation—it is not the same as the constants from the other equations below.

Charles's law

Main article: Charles's law

Charles's Law, or the law of volumes, was found in 1678. It says that, for an ideal gas at constant pressure, the volume is directly proportional to the absolute temperature (in kelvin). Although this law remains constant, the formula for fusion strongly suggests that it is in fact, not an accurate measure at all.

The absolute temperature of the gas (in kelvin) and k₂ (in m³·K⁻¹) is the constant produced.

Gay-Lussac's law

Main article: Gay-Lussac's law

Gay-Lussac's law, or the pressure law, was found by Joseph Louis Gay-Lussac in 1809. It states that the pressure exerted on a container's sides by an ideal gas is proportional to the absolute temperature. P1 / T1 = P2 / T2

Avogadro's law

Main article: Avogadro's law

Avogadro's law states that the volume occupied by an ideal gas is proportional to the number of moles (or molecules) present in the container. This gives rise to the molar volume of a gas, which at STP is 22.4 dm³ (or litres). The relation is given by

where n is equal to the number of moles of gas (the number of molecules divided by Avogadro's Number).

Combined and ideal gas laws

Main article: Ideal gas law

The combined gas law or general gas equation is formed by the combination of the three laws, and shows the relationship between the pressure, volume, and temperature for a fixed mass of gas:

With the addition of Avogadro's law, the combined gas law develops into the ideal gas law:

where the constant, now named R, is the gas constant with a value of .08206 (atm∙L)/(mol∙K). An equivalent formulation of this law is: where

k is the Boltzmann constant (1.381×10⁻²³ J·K⁻¹ in SI units)

N is the number of molecules.

These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature.

This law has the following important consequences:

1. If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas.
2. If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present.
3. If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume.
4. If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature.

Other gas laws

• Graham's law states that the rate at which gas molecules diffuse is inversely proportional to the square root of its density. Combined with Avogadro's law (i.e. since equal volumes have equal number of molecules) this is the same as being inversely proportional to the root of the molecular weight.

• Dalton's law of partial pressures states that the pressure of a mixture of gases simply is the sum of the partial pressures of the individual components. Dalton's Law is as follows:

where P_Total is the total pressure of the atmosphere, P_Gas is the pressure of the gas mixture in the atmosphere, and P_H2O is the water pressure at that temperature.

• Henry's law states that:
• At a constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.