# Gas Laws

## Introduction

In this section, we will be exploring gas behaviour.  The results of experimental observation and work are expressed here as 'laws' and are useful models to help us predict gas behaviour in the right circumstances.  We will develop the Kinetic molecular theory, which gives us insight into the molecular level reasons that explain gas behaviours.  Later in the unit, we will discover conditions under which the first gas laws do not predict the observed behaviour.  More advanced models are proposed to better describe gas behaviours under these conditions.

You will be expected to know each of the gas laws presented here, the conditions under which they work and do not work.  You will also be expected to be able to use these ideas and the equations within to solve problems both here in the Gas units but also later in other units of the course.

According to Avogadro, equal volumes of different (ideal) gases at the same temperature and pressure contain equal numbers of molecules (moles) of the different gases. This law explained Gay-Lussac Law that noted that volumes of gases reacting are related by small whole number ratios.

For example, in the reaction B

2H2(g) + O2(g) $\Rightarrow$ 2H2O(g)

2L        1L     $\Rightarrow$    2L

Volume Ratio of Reactants:Products :: 3L:2L

Avogadro's law assumes all atoms (molecules) are the same size. This assumption is only true at extremely small pressures. It's not bad for every-day atmospheric pressures.

Standard Temperatures and Pressures STP are defined to be
0�C and 100 kPa (=0.98692 atm) (NOTE 1)
one mole of Gas at STP occupies 22.7 L (NOTE 2)

There are slight variances in real gases at normal conditions but not much.

Example

What is the Molar mass of a gas of which 2.641 g occupies 705.6 mL @STP?

$\mathrm{0.7056\;L\; \left[ \frac{1\; mol}{22.7\;L} \right] \;=\; 0.0311\; mol}$

$\mathrm{\frac{2.640\; g}{0.0315 mol}\;=\;84.9\; g\; mol^{-1}}$

## Pressures of Gases

Pressure = force/area

 Diagram of inverted test tube in mercury.  The column of mercury of height, h, is supported by the air pressure.

Consider the following inverted tube filled with mercury such that its open end is immersed in a pool of mercury.

A standard rule in fluid dynamics states that the pressure of a liquid is constant at any given height. So the pressure inside and outside the tube at the height of the surface must be equal and opposite that of the air pressure. Thus, the pressure due to the weight of the column of mercury must be equal to the pressure of the air. By measuring the height of the column we have a direct measure of the air pressure.

Note that a gas at STP (T= 0ºC., P = 100 kPa) will support a column of mercury of height 750.0 mm. Hence, we some times take the short-cut way of expressing pressure as a height of a column of mercury that would be supported by the gas pressure being measured.  At STP, the pressure of an ideal gas is 750 mmHg. (NOTE 3)

Example

A sample of gas supports a column of mercury 680.2 mm high. What is the pressure of the gas in atm, and in kPa?

Example

What is the height of a column of water which can be support by a pressure of exactly 1 bar (=100 kPa = 750.0 mmHg) at 25ºC?

The density of water at that temperature is 0.997 g cm-3.

where h is the height of the column of liquid, d is its density, A is the area of the column and g is the acceleration due to gravity. We could look up the value for acceleration due to gravity and calculate h that way. However, we happen to know the height of a column of mercury that the same air pressure would support (750) therefore, let's just equate the pressure of water and the pressure of mercury as follows.

 PH2O = PHg dH2O × hH2O × g = dHg × hHg × g 0.997g/mL × hH2O = 13.6 g/mL × 750 mm hH2O = 13.6 × 750/.997 mm = 10.23 m

## Boyle's Law

Boyle measured the relationship between the pressure and the volume of a given sample of gas at fixed temperature. He found that a sample of gas compresses if the external pressure applied to it increases and that the product PV is constant.

Boyle's law, stated in mathematical terms for a gas whose pressure and volume is measured at two different pressure/volume states at a constant temperature is then,

 P1V1 = P2V2 .

Note: This law is only truly valid at infinitely low pressures. Standard pressures, however lead to a reasonable approximation for most gases.

Example:

A sample of air occupies a volume of 450.0 mL at 20 ºC and 1.00 bar (100 kPa). What will be the pressure of this gas if it is transferred to a 2.000 L bulb at the same temperature?

$P_1V_1 = P_2V_2$

$P_2 = \frac{P_1P_2}{V_2}= \frac{1.00\;\mathrm{bar} \times 0.4500\; \mathrm{L}}{2.000\; \mathrm{L}}=0.225\; \mathrm{bar} = 22.5\; \mathrm{kPa}$

A gas that obeys both Boyle's Law and Avogadro's law is an Ideal gas. The atoms (molecules) of an ideal gas are infinitely small such that they never collide with each other. They never even interact with each other in any way. Thus, the atoms/molecules of an ideal gas each behaves as if they were the only atom in the container. Simple experience tells us that this is not exactly true, even at room temperature (gas-phase reactions occur). It is a reasonably good assumption for most gases at or near STP. The assumption of ideality breaks down as the pressure in increased or as the temperature is lowered. It also breaks down sooner for particularly large or massive or polar gas molecules

## Charles' Law

The volume of any gas increases linearly with increasing temperature at constant pressure.

If we plot a graph of the volume of a sample of gas versus the temperature we get something that looks like the following:

 From the extrapolated line, we can determine the temperature at which an ideal gas would have a zero volume. Since ideal gases have infinitely small atoms the only contribution to the volume of a gas is the pressure exerted by the moving atoms bumping against the walls of the container. If no volume then there must be no kinetic energy left. Thus, absolute zero is the temperature at which all kinetic energy (motion) has been removed. NOTE: This does not mean all energy has been removed, merely all kinetic energy.

To make the V versus T function simpler, we should use the absolute temperature scale (Kelvin)

TC = TK + 273.15.

(Now we get simply a straight line plot with an intercept of zero and a slope of some constant k).

To avoid the need to know k, we use ratios. The ratio of V to T of an ideal gas at constant pressure is constant over all temperatures. Or...

 $\frac{V_1}{T_1}\;=\;\frac{V_2}{T_2}$

Example

A sample of gas occupies 400.0 mL at 25.00ºC and 1 bar pressure. What volume will it occupy at 200.00ºC at the same P?

$\frac{V_1}{T_1}\;=\;\frac{V_2}{T_2}\;\Rightarrow\; V_2\;=\; V_1\;\frac{T_2}{T_1}$

$V_2\;=\;400.0\;mL\frac{(200.00+273.15)K}{(25.00+273.15)K}\;=\;634.8\;mL$

## Ideal Gas Law

If we take the three gas laws we've studied so far, we can combine them into a single law called the Ideal Gas law. This law covers the relationship between temperature, pressure, volume and number of moles of an Ideal gas.

 Avogadro's Law V = k1n |T,P Boyle's Law V = k2/P |T,n Charles' Law V = k3T |n,P

After some consideration and algebra, we arrive at

V = koverall nT/P

where koverall turns out to be the Ideal gas constant (or universal gas constant)

R = 8.314 462 J mol-1K-1.

We're more familiar with the equation written as:

 PV = nRT

This is the Ideal Gas Law.

Example:

A sample of butane (C4H10) of mass 3.728 g is placed in an evacuated bulb of volume 489 mL at 25.ºC. What is its pressure?

PV = nRT

$n=3.728 g \times \frac{1 \;\mathrm{mol}}{(4 \times 12.011 + 10 \times 1.008) \mathrm{g/mol}}$

$n = 6.414\times10^{-2}\; \mathrm{mol}$

$P=\frac{nRT}{V}= \frac{0.06414\; \mathrm{mol} \times 8.31451\; \mathrm{J/mol K} \times 298\;\mathrm{K}}{489\; \mathrm{cm}^3\left( 1m/100cm\right)^3}$

$P = 319\; \mathrm{kPa}$

NOTE: in this case, we needed to be sure that all units were SI units. Then the pressure simply worked out to be the SI unit of pressure.

There are many kinds of calculations that involve the Ideal Gas Law.  Many of them involve far more than simply substituting values into the equation and getting an answer.  Let's try one example where we use the Ideal gas law to set up the problem but we don't actually complete a calculation using the Ideal Gas Equation.

Example:

A bulb is filled with H2 gas at a temperature T. The pressure is 756 mmHg. A portion of the gas is transferred to a flask and at 100 kPa, (NOTE: 100 kPA = 750.0 mmHg) occupies 40.0 mL at the same temperature T. The pressure in the original bulb drops to 625 mmHg at temp. T. What is the volume of the bulb?

This is a many step problem and there is no way you can expect to be able to 'know' how to solve it at first glance. You do, however have enough knowledge to solve it now. You just have to put down what you know and see how it all fits.

PV = nRT

let:

n = total # moles of H2(g)
V = Volume of bulb

 n = n1 + n2 (1)

where n1 is the number of moles removed to the flask and n2 is the number of moles remaining behind.

 Initial: 756 mmHg × V = nRT (2) transfered: 750 mmHg × 40.0 mL = n1RT (3) Remaining: 625 mmHg × V = n2RT (4)

At this point, we have 4 equations in what seems to be 5 unknowns, n, n1, n2, V and T.  However, T is a constant in this process (as well as R) and we will be able to eliminate both R and T through some simple math.  If we actually needed to determine T then we would need a 5th equation. but since we are not interested in it's value, we only need 4 equations, so long as we can eliminate it some way, which we can.

Now substitute (1) into (2)

756mmHg × V = (n1 + n2)RT = n1 RT + n2 RT

Now sub in (3) and (4) (thus eliminating RT from the picture)

756mmHg × V = 750mmHg × 40.0mL + 625mmHg × V

Cancel out the units mmHg (NOTE 4 ) and gather the known terms to the right-hand side.

$V\;=\;\frac{750\;mm\;Hg\;\times\;40.0\;mL}{756\;mm\;Hg\;-\;625\;mm\;Hg}\;=\;232\;mL$

## Dalton's Law of Partial Pressures

If more than one gas occupy a single container then the number of moles of each gas in proportional to the pressure of each gas (the gas' partial pressure) and the total pressure is equal to the sum of all the partial pressures. In like manner, the total number of moles of gas is equal to the sum of the numbers of moles of the individual gases.

This actually follows from the Ideal Gas law PV = nRT .

For Dalton's Law to work all gases in the mixture must behave like ideal gases under the conditions studied.

$P_1V\; =\; n_1RT\;\;\Rightarrow\;\;P_1\;=\;\frac{n_1 RT}{V}$

and

$P_2V\; =\; n_2RT\;\;\Rightarrow\;\;P_1\;=\;\frac{n_2 RT}{V}$

Dalton's Law says the total pressure is simply the sum of the individual gas pressures (a.k.a. partial pressures) so we substitute for P1 and P2.

$P_T\; =\; P_1 \;+\; P_2\; =\; \frac{n_1 RT}{V}\; +\; \frac{n_2 RT}{V}\; =\;\frac{(n_1\;+\; n_2)RT}{V} \;=\; \frac{n_T RT}{V}$

This allows us to make a few quick calculations in our head. Useful in certain gas-law type problems. Let's take, for example, the atmosphere.

The air is made up of

78.08 % N2 (mole %)
20.95 % O2
0.934% Ar + ...

This quickly translates into (for a 100 mole sample of air)

78.08 mol N2
20.95 mol O2
0.934 mol Ar + ...

or, if we have 100 mmHg total pressure, then the partial pressures are

78.08 mmHg of N2
20.95 mmHg of O2
0.934 mmHg of Ar + ...

Consider an experiment where we're collecting a certain gas by bubbling it into an inverted glass jar initially filled with water.

The partial pressure of water, pH2O, in this case is actually the vapour pressure, P*H2O, and can be found in any one of several tables of thermodynamic data.

There are many other situations in which we must consider the pressure of more than one gas at at time.  Consider, for example:

A closed bulb contains 0.0100 mol of He(g) and a sample of solid white ammonium chloride (NH4Cl). The pressure of the Helium at 27ºC is 114 mmHg.

The bulb is then heated to 327ºC and the ammonium chloride decomposes into ammonia gas and hydrogen chloride gas. The final pressure is 908 mmHg.

1. What is pHCl @ 327ºC ?
2. How many grams of ammonium chloride were in the bulb initially?

a.  We sum up the partial pressures.

PT = pHe + pNH3 + pHCl = 908 mmHg (after the reaction @ 327ºC)

According to the ideal gas law,

$\frac{P}{T}\;=\;\frac{nR}{V}.$

Since, for the He, n and V are constant then we can use the P/T constant ratio to determine He pressure at 327ºC.

$p_{He}(327)\;=\;p_{He}(27)\times\frac{(327+273)\mathrm{K}}{(27+273)\mathrm{K}}$

$p_{He}(327)\;=\;114\;\mbox{mm Hg}\;\frac{600\;\mathrm{K}}{300\;\mathrm{K}}\;=\;228\;\mbox{mm Hg}$

The balanced chemical reaction for the decomposition of ammonium chloride is

NH4Cl(s) $\rightarrow$ NH3(g) + HCl(g)                1:1 ratio. therefore:

pNH3 = pHCl = X

PT = 228 mm Hg + 2X = 908 mHg

X = 340 mmHg = pHCl

b. We start by writing out the number of moles of HCl and of He since we can use this in Dalton's Law.

$n_{\mathrm{HCl}} = p_{\mathrm{HCl}}V/RT$

(We don't know V or T so we need at least one more equation)

$n_{\mathrm{He}} = p_{\mathrm{He}}V/RT$

Now divide these two equations one into the other to eliminate the V/RT .

$\frac{n_{\mathrm{HCl}}}{n_{\mathrm{He}}}\;=\;\frac{p_{\mathrm{HCl}}}{p_{\mathrm{He}}}$

$n_{\mathrm{HCl}}\;=\;n_{\mathrm{He}}\frac{p_{\mathrm{HCl}}}{p_{\mathrm{He}}}\;=\;0.0100\;\mbox{mol}\;\frac{340\;\mbox{mm Hg}}{228\;\mbox{mm Hg}}\;=\;0.0149\;\mbox{mol}\;\mathrm{HCl}$

Now, we convert the moles of HCl into g of NH4Cl using an extended fraction.  To do this, we need two conversion factors.  The first is the mole ratio from the balanced chemical equation.  The second is the molar mass of ammonium chloride.

 0.0149 mol HCl 1 NH4Cl 53.49 g NH4Cl = 0.798 g NH4Cl 1 HCl 1 mol NH4Cl

## The Kinetic Molecular Theory

The document that fits here is in a separate file.  Click Here to view it.

## The van der Waals equation

We've already discussed the postulates behind the ideal gas law but let's recap here.

Ideal gases have molecules:

• that are infinitely small in size,
• that don't interact with each other.

Is there a way to take into account the fact that these two assumptions are not always valid?  To account for these two properties of the material which the Ideal gas law ignores, we need two new parameters. In 1873, J. H. van der Waals proposed a new equation which attempted to account for the size of the atoms and their interactions. The parameterized phenomenological equation contained two adjustable parameters a and b. These two parameters are varied to fit the equation results to the experimental properties of the gas in question.

The van der Waals equation is

$\left[ P+\frac{n^2 a}{V^2} \right] (V-nb)\;=\;nRT$

or by dividing both sides by n, we get an expression that uses molar volume,

$\left[P+\frac{a}{V^2_m}(V_m-b)\;=\;RT \right]$

where Vm = V/n = molar volume of the gas.  For most gases, the values of a and b are quite small (See table of values).

To try to understand the meaning of the two parameters, we can rearrange the vdW equation to have pressure on the left hand side and then compare this equation with the ideal gas law.

$P\;=\; \frac{nRT}{V}$

The vdW equation rearranges to be

$P\;=\;\frac{RT}{V_m-b}-\frac{a}{V^2_m}$

Recall that Vm is the molar volume. As a gas is compressed, the molar volume is reduced, which makes the relative size of the parameters a and b larger. Consider the situation where the the molar volume is large (say, near standard temp and at or below standard pressure) then the second term $\frac{a}{V^2_m}$ will be quite small (because a is quite small compared to Vm). Similarly, the first term reduces the the ideal gas law, since Vm is much larger than b, the denominator of the first term can be assumed to be identical to Vm with a high degree of accuracy. In other words, if the molar volume is large, the assumptions made in the ideal gas law are valid. As the molar volume becomes smaller, then the corrections made in the vdW equation become important.

The parameter a serves as a correction to the pressure of the gas as a result of intermolecular attractive forces. As the molar volume is reduced, the molecules are forced closer together and the interactions become significant.

 Figure: The diagram on the left shows a low density (high molar volume) case where the atom, which is about to strike the wall (apply pressure) has relatively few attractive forces with other gas molecules because of large distances between them. On the right, the molar volume has been reduced and the resulting larger number of intermolecular forces acting on the molecules as they collide with the wall effectively reduce the force of the collision (pressure is lower than predicted by Ideal Gas law).

• The parameter b serves to correct the molar volume of the gas. An ideal gas has molecules that have no size. We need to correct for this assumption. If the molecules have a non-zero finite size then they will have, in effect less volume in which to travel since some of the volume of the container is now taken up by the molecules themselves. At low densities (high molar volumes) this is not very significant since the gas molecules are so far apart that they rarely interact anyway. As the gas is compressed, the molar volume of the gas is reduced to the point where the volume of the molecules becomes an important factor to consider.
 Figure: The two cylinders contain the same amount of gas molecules (black dots) On the left, the molar volume is larger and the amount of free space for the molecules to move around in is larger than the volume of the molecules themselves. On the right, the cylinder has been compressed (smaller molar volume) and the gas molecules have much less free space to move about in. Now, the volume of the molecules themselves take up a significant portion of the volume of the cylinder, effectively reducing the volume available to any one molecule.

For now, it is important to note that the parameters a and b are related to intrinsic properties of the gas itself and are therefore different for each gas.  The following figure shows isotherms for three different gases as calculated using the van der Waals equation shown with T = 586 K.  Curve 2 (black) is an ideal gas with parameters a = b = 0.  Curve 1 (blue) is a gas with a vary small value for a but a moderate value for b.  Note that at this temperature, there is no region where the gas pressure is smaller than a similarly conditioned ideal gas.  Curve 3 (red) is a gas with a fairly large value of a and a moderate value for b.  Note that for this gas, there are some conditions where the calculated pressure is lower than that of a similarly conditioned ideal gas.  Parameter a is a measure of the attractive forces.  Hence, the blue has has very little attractive forces but some molecular volume (non-zero b) so the pressure only goes up as the molar volume goes down.  The red gas has a slightly larger molecular volume than the blue gas and much larger attractive forces so as the molar volume goes down, you see both effects.  As the molar volume is decreased, the attractive forces begin to dominate.  Eventually at very small molar volume, the repulsive forces (molecular volume) takes over and you see a rapid rise in the pressure for that gas.

We have an Excel spreadsheet that can be used to explore the properties of a gas at non ideal conditions as modeled by the van der Waals' equation.

Through the use of this sheet, you can see that as parameter a is increased, the measured pressure is decreased, compared to an ideal gas (= 0) pressure at the same conditions.  Similarly, you can see that as b  is increased, the pressure deviates positively from the ideal gas pressure.

$z\;=\;\frac{V_m}{V^o_m}$