Chapter 1

Ideal Law

Real Gas

Condensed Matter-solid and liquid

Triple Point

Van der Waals of State Equation

PHY351

Gas Law

Gases are a state of matter characterized by two properties which are lack of exact volume and lack of exact shape.

There are three properties of gases: Volume, V Pressure, P Temperature, T

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The Gas Laws and Absolute Temperature

The relationship between the volume, pressure, temperature, and mass of a gas is called an equation of state.

The gases law that we will cover for this chapter including: Boyle`s law

Charles`s law

Pressure law or Gay-Lussac law

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Boyle’s Law

The volume of a gas depends on the pressure exerted on it. When you pump up a bicycle tire, you push down on a handle that squeezes the gas inside the pump. You can squeeze a balloon and reduce its size (the volume it occupies).

In general, the greater the pressure exerted on a gas, the less its volume and vice versa.

This relationship is true only if the temperature of the gas remains constant while the pressure is changed.

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Pressure is inversely proportional to the volume.

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Boyle‟s law definition:

..’The volume of a given amount of gas is inversely

proportional to the absolute pressure as long as the

temperature is constant’..

We can see that the data fit into a pattern called a hyperbola.

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If, however we plot pressure against 1/volumewe get a linear (straight line) graph.

PV = constant

Thus;

P1V1 = P2V2

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Charles‟s law definition:

..’ the volume of a given amount of gas is directly

proportional to the absolute temperature (Kelvin) when the

pressure is kept constant’..

Charles’s law

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V/T = constant

Thus;

V1/T1 = V2/T2

By extrapolating, the volume becomes zero at −273.15°C; this temperature is called absolute zero (refer figure).

Therefore, absolute temperature:

T(K) = T(C) + 273.159

Definition:

..’The absolute pressure of a given amount of gas is directly proportional to the absolute temperature

(K) when the volume is kept constant’..

P/T = constant

Thus; P1/T1 = P2/T2

Gay-Lussacs’ law

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Absolute Temperature

Known as the Kelvin scale. Degrees are same size as Celcius scale.

Absolute zero = 0°C = 273.15 K

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At standard atmospheric pressure :

Room temperature is about 293 K (20°C)

Standard temperature is 273 K

Water freezes at 273.15 K and boils at 373.15 K.

T (K) = T (°C) + 273.15

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Note:If any gas could be cooled to -273°C, the volume would

be zero.

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This three gas laws can be combined to produced a single more general relation between absolute pressure, volume and absolute temperature; that is:

Where PV/T = constant

Thus;

P1V1/T1 = P2V2/T2

The Ideal Gas Law

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Now we are looking to a simple experiment where the balloon is blown up at a constant pressure and temperature (figure below).

It is found that, the volume, V of a gas increases in direct proportion to the mass, m of a gas present:

V m

Hence we can write:

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A mole (mol) is defined as the number of grams of a substance that is numerically equal to the molecular mass of the substance.

Example:

i. 1 mol H2 has a mass of 2 g.

ii. 1 mol Ne has a mass of 20 g.

iii. 1 mol CO2 has a mass of 44 g; where the molecular mass for CO2 = [12 + (2 x 16)] = 44 g/mol

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Where the number of moles in a certain mass of material is given as:

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From proportion the equation for IDEAL GASlaw can be written as:

Where n is the number of moles and R is the universal gas constant and its value is given as:

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The ideal gas law often refers to “standard conditions” or standard temperature and pressure (STP). Where at STP:

T = 273 K

P = 1.00 atm

= 1.013 x 105 N/m2

= 101.3 kPa

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Note:1 mol STP gas has:i. Volume = 22.4Lii. No. of molecule/moles = 6.023 x 1023 molecules

Note:

When using and dealing with all this three gas laws and also ideal gas law; the temperature must in Kelvin (K) and the pressure must always be absolute pressure, not gauge pressure.

Absolute pressure = gauge pressure + atmospheric pressure

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1. Determine the volume of 1.00 mol of any gas, assuming it behaves like an ideal gas, at STP.

(ans: 0.0224 m3)

2. An automobile tire is filled to a gauge pressure of 200 kPa at 10°C. After a drive of 100 km, the temperature within the tire rises to 40°C. What is the pressure within

the tire now?

(ans: 333 kPa)

Exercises:

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Since the gas constant is universal, the number of molecules in one mole is the SAME for all gases. That number is called Avogadro‟s number:

The number of molecules in a gas is the number of moles times Avogadro‟s number:

Ideal Gas Law in Terms of Molecules: Avogadro’s Number

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Therefore we can write:

Where k is called Boltzmann‟s constant

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Kinetic Theory and the Molecular Interpretation of Temperature

The force exerted on the wall by the collision of one molecule is:

Then the force due to all molecules colliding with that wall is:

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Figure 13.16:(a)Molecules of a gas moving about

in a rectangular container.

(a) Arrows indicate the momentum of one molecules as it rebounds from the end wall.

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The averages of the squares of the speeds in all three directions are equal:

So the pressure is:

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Rewriting;

So;

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The average translational kinetic energy of the molecules in an ideal gas is directly proportional to the temperature of the gas.

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We can invert this to find the average speed of molecules in a gas as a function of temperature:

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Distribution of Molecular Speeds

These two graphs show the distribution of speeds of molecules in a gas, as derived by Maxwell. The most probable speed, vP, is not quite the same as the rms speed.

Note:vrms is not at the peak of the curve because the curve is skewed to the right (not symmetrical).vp is called the „most probable speed.

Figure 13.17: Distribution of speeds of molecules in an ideal gas.

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As expected, the curves shift to the right with temperature.

Figure 13.18: Distribution of molecular speeds for two different temperature

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The ideal gas law equation is given as;

The term of “ideal” refers to characteristics/behavior of gas where at “ideal”, the pressure of gas is too high and the temperature of gas close/near the liquefaction point (boiling point).

Real Gases

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However, at pressures less than an atmosphere or so (not too high), and when temperature is not close to the boiling point of the gas, it is refers to behavior of real gases.

Figure below is the curve of P vs V at temperature constant for ideal gas (Boyle`s law).

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And figure below is the curves represent the behavior of the gas at different temperatures (not constant) for real gases. Where TA TB TC TD

It is found that, the cooler (temperature decrease @ farther from boiling point) it gets, the further the gas is from ideal.

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Below the critical temperature (the gas can liquefy if the pressure is sufficient; above it, no amount of pressure will suffice):

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The dashed curve A` and B` represents the behavior of a gas as predicted by the ideal gas law (Boyle`s law) for several different values of the temperature.

We see that, the behavior of gas deviates even more from the curves predicted by ideal gas law (curves A and B), and the deviation is greater when the gas is closer to liquid-vapor region (curve C and D).

Figure 13.19: PV diagram for real substance

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In curve D, the gas becomes liquid; it begins condensing at (b) and is entirely liquid at (a).

Curve C represent the behavior of the substance at its critical temperature, and the point (c) is called the critical point.

At temperature less than the critical temperature, a gas will change to the liquid phase if sufficient pressure is applied.

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The behavior of a substance can be diagrammed not only on a PV diagram but also on a PT diagram.

A PT diagram is called a phase diagram; it shows all

three phases of matter:

The solid-liquid transition (in equilibrium) is melting or freezing

The liquid-vapor transition is boiling or condensing

The solid-vapor transition is sublimation. Where sublimation refers to the process whereby at low pressures a solid changes directly into the vapor phase without passing through the liquid phase.

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The intersection of the three curves is called the triple point. Where it is only at triple point that the three phases can exist together in equilibrium.

Figure 13.20: Phase diagram of water.

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Figure 13.21: Phase diagram of carbon dioxide.

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Example

23. CO2exists in what phase when the pressure is 30 atm and the temperature is 30°C (Fig. 18–6)?

24. (a) At atmospheric pressure, in what phases can exist?

(b) For what range of pressures and temperatures can be a liquid? Refer to Fig. 18–6.

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25. Water is in which phase when the pressure is 0.01 atm and the temperature is

(a) 90°C

(b) -20oC

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26. You have a sample of water and are able to control temperature and pressure arbitrarily.

(a) Using Fig. 18–5, describe the phase changes you would see if you started at a temperature of 85°C, a pressure of 180 atm, and decreased the pressure down to 0.004 atm while keeping the temperature fixed.

(b) Repeat part (a) with the temperature at 0.0°C. Assume that you held the system at the starting conditions long enough for the system to stabilize before making further changes.

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Van Der Waals of State Equation

To get a more realistic model of a gas, we include the finite size of the molecules and the range of the intermolecular force beyond the size of the molecule.

Molecules of radius, r colliding.

The van der Waals equation is:

Where;

P = pressure (atm) V = volume (liter)

n = number of moles (mol) T = temperature (Kelvin,K)

R = ideal gas constant a and b = constants

The PV diagram for a Van der Waals gas fits most

experimental data quite well.

PV diagram for a van der Waals gas, shown for four different temperature.

For TA, TB and TC (TC is chosen equal to the critical temperature), the curves fit the experimental data very well for most gases.

The curves labeled TD, a temperature below the critical point, passes through the liquid-vapor region.

The maximum (poin b) and minimum (point d) would seem to be artifacts, since we usually see constant pressure, as imdicated by the horizontal dashed line.

However, for very pure supersaturated vapors or supercooled liquids, the sections ab and ed respectively, have been observed.

The section bd would be unstable and has not been observed.

Example

41.For oxygen gas, the van der Waals equation of state achieves its best fit for and Determine the pressure in 1.0 mol of the gas at 0°C if its volume is 0.70 L, calculated using

(a) the van der Waals equation

(b) the ideal gas law.

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Reference

Giancoli, DC., (2005). Physics 6th Edition. New Jersey: Prentice Hall.

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