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Gas Laws. M. L. Watson. Ideal Gas Law.

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### Gas Laws

M. L. Watson

An ideal gas is defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces. One can visualize it as a collection of perfectly hard spheres which collide but which otherwise do not interact with each other. In such a gas, all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature. An ideal gas can be characterized by three state variables: absolute pressure (P), volume (V), and absolute temperature (T). The relationship between them may be deduced from kinetic theory and is called the

Where:

n = number of moles

R = universal gas constant = 8.3145 J/mol K

N = number of molecules

k = Boltzmann constant = 1.38066 x 10-23 J/K = 8.617385 x 10-5 eV/K

k = R/NA

NA = Avogadro's number = 6.0221 x 1023

Ideal Gas Law

PV=nRTZ

n = Mass

R = Universal gas constant

T = Temperature

Z = Supercompressability

(P1V1/T1)Z1=(P2V2/T2)Z2

Ideal Gas Law

The ideal gas law can be viewed as arising from the kinetic pressure of gas molecules colliding with the walls of a container in accordance with Newton's laws. But there is also a statistical element in the determination of the average kinetic energy of those molecules. The temperature is taken to be proportional to this average kinetic energy; this invokes the idea of kinetic temperature. One mole of an ideal gas at STP occupies 22.4 liters.

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 = nRTwhere P=pressure, V=volume, n=moles of gas, T=temperature, R is the gas constant which is dependent on the units of pressure, temperature and volumeR = 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
- Gases consist of small particles (molecules) which are in continuous random motion
- The volume of the molecules present is negligible compared to the total volume occupied by the gas
- Intermolecular forces are negligible
- Pressure is due to the gas molecules colliding with the walls of the container
- Real Gases deviate from Ideal Gas Behaviour because
- at low temperatures the gas molecules have less kinetic energy (move around less) so they do attract each other
- 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.

There are many applications to the Ideal Gas Law Equation

How can the ideal gas law be applied in dealing with how gases behave?

- PV = nRT
- Used to derive the individual ideal gas laws:
- For two sets of conditions: initial and final set of conditions:
- P1V1 = n1RT1 and P2V2 = n2RT2
- Solving for R in both equations gives:
- R = P1V1 / n1T1 and R = P2V2 / n2T2
- Since they are equal to the same constant, R, they are equal to each other:
- P1V1 / n1T1 = P2V2 / n2T2
- For the Volume Pressure relationship (ie: Boyle's Law):
- n1 = n2 and T1 = T2 therefore the n's and T's cancel in the above expression resulting in the following simplification:
- P1V1 = P2V2 (mathematical expression of Boyle's Law)
- For the Volume Temperature relationship (ie: Charles's Law):
- n1 = n2 and P1 = P2 therefore the n's and the P's cancel in the original expression resulting in the following simplification:
- V1 / T1 = V2 / T2 (mathematical expression of Charles's Law)
- For the Pressure Temperature Relationship (ie: Gay-Lussac's Law):
- n1 = n2 and V1 = V2 therefore the n's and the V's cancel in the above original expression:
- P1 / T1 = P2 / T2 (math expression of Gay Lussac's Law)
- For the Volume mole relationship (Avagadro's Law)
- P1 = P2 and T1 = T2 therefore the P's and T's cancel in the above original expression:
- V1 / n1 = V2 / n2 (math expression for Avagadro's Law)
- Used to solve single set of conditions type of gas problems where there is no observable change in the four variables of a gas sample. Knowing three of the four variables allows you to determine the fourth variable. Since the universal Gas Law constant, R, is involved in the computation of these kinds of problems, then the value of R will set the units for the variables.

Ideal Gas Law w/Constraints

For the purpose of calculations, it is convenient to place the ideal gas law in the form:

where the subscripts i and f refer to the initial and final states of some process. If the temperature is constrained to be constant, this becomes:

which is referred to as Boyle's Law.

If the pressure is constant, then the ideal gas law takes the form

which has been historically called Charles' Law. It is appropriate for experiments performed in the presence of a constant atmospheric pressure.

Boyle’s Law

- At constant temperature, the volume of a given quantity of gas is inversely proportional to its pressure : V 1/PSo at constant temperature, if the volume of a gas is doubled, its pressure is halved.ORAt constant temperature for a given quantity of gas, the product of its volume and its pressure is a constant : PV = constant, PV = k
- At constant temperature for a given quantity of gas : PiVi = PfVfwhere Pi is the initial (original) pressure, Vi is its initial (original) volume, Pf is its final pressure, Vf is its final volumePi and Pf must be in the same units of measurement (eg, both in atmospheres), Vi and Vf must be in the same units of measurement (eg, both in litres).
- All gases approximate Boyle's Law at high temperatures and low pressures. A hypothetical gas which obeys Boyle's Law at all temperatures and pressures is called an Ideal Gas. A Real Gas is one which approaches Boyle's Law behaviour as the temperature is raised or the pressure lowered.

Boyle’s Law

P1V1=P2V2

Charles Law

- At constant pressure, the volume of a given quantity of gas is directly proportional to the absolute temperature : V T (in Kelvin)So at constant pressure, if the temperature (K) is doubled, the volume of gas is also doubled.ORAt constant pressure for a given quantity of gas, the ratio of its volume and the absolute temperature is a constant : V/T = constant, V/T = k
- At constant pressure for a given quantity of gas : Vi/Ti = Vf/Tfwhere Vi is the initial (original) volume, Ti is its initial (original) temperature (in Kelvin), Vf is its final volme, Tf is its final tempeature (in Kelvin)Vi and Vf must be in the same units of measurement (eg, both in litres), Ti and Tf must be in Kelvin NOT celsius.temperature in kelvin = temperature in celsius + 273 (approximately)
- All gases approximate Charles' Law at high temperatures and low pressures. A hypothetical gas which obeys Charles' Law at all temperatures and pressures is called an Ideal Gas. A Real Gas is one which approaches Charles' Law as the temperature is raised or the pressure lowered.As a Real Gas is cooled at constant pressure from a point well above its condensation point, its volume begins to increase linearly. As the temperature approaches the gases condensation point, the line begins to curve (usually downward) so there is a marked deviation from Ideal Gas behaviour close to the condensation point. Once the gas condenses to a liquid it is no longer a gas and so does not obey Charles' Law at all.Absolute zero (0K, -273oC approximately) is the temperature at which the volume of a gas would become zero if it did not condense and if it behaved ideally down to that temperature.

Supercompressability

Why? Molecular Squish

- under 1% for pressures under 1500 PSI

- different for each gas mixture

- similar to alcohol & water or sand & gravel

Theoretical

Actual

Isentropic Expansion

Thermodynamics!

P1V1k=P2V2k

k=(Cp/Cv)

k=Ratio of specific heat at a constant pressure

to the specific heat at a constant volume

K=1.3 for natural gas

Specific Heat = kJ/kg

All the possible states of an ideal gas can be represented by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

Molecular Constants by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

In the kinetic theory of gases, there are certain constants which constrain the ceaseless molecular activity. A given volume V of any ideal gas will have the same number of molecules. The mass of the gas will then be proportional to the molecular mass. A convenient standard quantity is the mole, the mass of gas in grams equal to the molecular mass in amu. Avogadro's number is the number of molecules in a mole of any molecular substance.

The average translational kinetic energy of any kind of molecule in an ideal gas is given by

State Variables by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

A state variable is a precisely measurable physical property which characterizes the state of a system, independently of how the system was brought to that state. It must be inherently single-valued to characterize a state. For example in the heat-work example, the final state is characterized by a specific temperature (a state variable) regardless of whether it was brought to that state by heating, or by having work done on it, or both.

Common examples of state variables are the pressure P, volume V, and temperature T. In the ideal gas law, the state of n moles of gas is precisely determined by these three state variables. If a property, e.g., enthalpy H, is defined as a combination of other state variables, then it too is a state variable. Enthalpy is one of the four "thermodynamic potentials", and the other three, internal energy U, Helmholtz free energy F and Gibbs free energy G are also state variables. The entropy S is also a state variable.

Some texts just use the term "thermodynamic variable" instead of the description "state variable".

The Mole by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

A mole (abbreviated mol) of a pure substance is a mass of the material in grams that is numerically equal to the molecular mass in atomic mass units (amu). A mole of any material will contain Avogadro's number of molecules. For example, carbon has an atomic mass of exactly 12.0 atomic mass units -- a mole of carbon is therefore 12 grams. For an isotope of a pure element, the mass number A is approximately equal to the mass in amu. The accurate masses of pure elements with their normal isotopic concentrations can be obtained from the periodic table.

One mole of an ideal gas will occupy a volume of 22.4 liters at STP (Standard Temperature and Pressure, 0°C and one atmosphere pressure).

Avogadro's number

Standard Temperature and Pressure by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

STP is used widely as a standard reference point for expression of the properties and processes of ideal gases. The standard temperature is the freezing point of water and the standard pressure is one standard atmosphere. These can be quantified as follows:

Pressure ( P ) is the ratio of the force applied to a surface (F) to the surface area ( A ).

P = F / A

Standard temperature: 0°C = 273.15 K = 32 F

Standard pressure = 1 atmosphere = 760 mmHg = 101.3 kPa =14.696 psi

Standard volume of 1 mole of an ideal gas at STP: 22.4 liters

Gauge Pressure by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

Does the flat tire on your automobile have zero air pressure? If it is completely flat, it still has the atmospheric pressure air in it. To be sure, it has zero useful pressure in it, and your tire gauge would read zero pounds per square inch. Most gauges read the excess of pressure over atmospheric pressure and this excess is called "gauge pressure". While a useful measurement for many practical purposes, it must be converted to absolute pressure for applications like the ideal gas law.

Since a partial vacuum will be below atmospheric pressure, the phrase "negative pressure" is often used. Certainly there is no such thing as a negative absolute pressure, but small decreases in pressure are commonly used to entrain fluids in sprayers, in carburetors for automobiles, and many other applications. In the case of respiration, we say that the lungs produce a negative pressure of about -4 mmHg to take in air, which of course means a 4 mmHg decrease from the surrounding atmospheric pressure.

Example by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

- Here is a sample problem involving a single set of conditions:
- A sample of chlorine gas weighs 1.31 g at STP. Calculate the volume this sample of chlorine would occupy under the following new conditions.
- 3.20atm and 0.00 C
- Here are the steps that will lead to an answer:
- Calculate the moles of Cl2 from 1.31 grams Cl2 using the formula weight of 2 x at weight of Cl

- 1.31 grams Cl2 X 1 mole Cl2 / 71 grams Cl2 = 0.0184 moles Cl2
- Check the temperature and convert to Kelvin if necessary:

- K = C + 273 = 0.00 + 273 = 273 K
- Check the pressure given and convert to atmospheres unit

- Pressure is already in atmospheres, 3.20 atm
- Use the value of R = 0.0821 liter-atm/mole-K
- Using the PV = nRT plug in the moles, temperature, pressure, and R and solve for the Volume in liters

- V = nRT / P = (0.0184 moles) ( 0.0821 liter-atm / mol-K) (273 K) / 3.20 atm = 0.129 liters

Example by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

- Let's let you try one:
- A sample of chlorine gas weighs 1.31 g at STP. Calculate the volume this sample of chlorine would occupy under the following new conditions.
- 760 torr and -23.0*C
- Use the Ideal Gas Law Equation to solve Gas stoichiometry problems involving conversion of moles of gas to volume or the conversion of a given volume of a gas sample under a stated temperature and pressure to moles or mass of another component in the process under a stated temperature and pressure.
- Use of Ideal Gas Law Equation to determine the density of a gas.
- PV = nRT if n = mass of gas / molecular mass
- then PV = (mass of gas/molecular mass) R T Solving for molecular mass
- molecular mass = (mass of gas) R T / P V
- Since Density = mass / V Then:
- molecular mass = D R T / P
- and solving for Density: D = (molecular mass) (P) / R T
- Using the equation to solve for Partial Pressure of a known amount of gas in a gas mixture:
- P1 = n1 R T / V and P2 = n2 R T / V

Example by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

- 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 kPan = 0.050 molT = 400KV = ? LR = 8.314 J K-1 mol-1 202.6V=0.050x8.314x400202.6 V = 166.28V = 166.28 ÷ 202.6V = 0.821 L (821mL)

Notes by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

Notes by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

Notes by a PvT surface as illustrated. The behavior when any one of the three state variables is held constant is also shown.

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