Chapter 1   Properties of Gases

 

-        The properties of gases

 

gas - the simplest state of matter (fills any container it occupies)

        - perfect gas can be pictured as a collection of molecules in continuous random motion, with speeds that increase with Temp >.  Different than a liq. in that, except during collisions, the molecules of the gas are widely separated from one another and move in paths that are largely unaffected by intermolecular forces.

 

The physical state of a substance is defined by its physical properties, and two samples of a substance that have the same physical properties are in the same state.

 

The state of a pure gas is specified by giving values of V, n, P, & T.

                p = f (T,V,n)  general form of equation of state

 

Experimentally determination of 3 of the 4 properties fixes the fourth and the properties of a simple system.  f (P, V, T, n ) = 0(All 4 variables are not independent of each other)

Mathematical form for the perfect or ideal gas:

R is the gas constant.


 

 


The perfect gas obeys the perfect gas equation (mathematical expression)

 

 



Fig. 1.9

Pressure is defined as force divided by the area to which the force is applied (force per unit area). Normally reported in Pascals, Pa, where 1 Pa = 1 N m-2 , & 1 atm = 1.01325 x 105 Pa.

 

A pressure of 105 Pa (1 bar) is the standard pressure for reporting data, denoted  Pø.

 

 

 

Calculating Pressure – Example

 

Dibutyl phthalate is commonly used as a liquid in manometers.

 

What pressure is equivalent to a centimeter of this liquid?

r for dibutyl phthalate is = 1.0465 g/cc, g = 9.8067 m/s2

 

 

 

P = f/A = mg/A = mgh/V = rgh

 

   f=ma               V=hxA     r =m/V


 


Calculate the pressure exerted by a mass of 1 kg pressing through the point of a pin of area 1 x 10-2 mm2 @ Earth’s surface?

 

P = F/A = mg/A =1kg (9.81m/s2)/(.01mm2)(1000mm/1m)2

 

P = 980,000,000 Pa = 9.7 x 103  atm.

 

Pressure exerted by a column of liquid is given by:

P = rgh ,     (F = mg = rAhg)

 

Measurement of pressure of a gas inside a container can be done with a manometer.


Fig. 1.2     a) P = rgh          b) P  = Pex + rgh

 

 


Temperature

 

The concept of temperature springs from the observation that a change in physical state (ie V) can occur when two objects are in contact with one another.

 

Temperature is a property that tells us whether two objects would be in thermal equilibrium if they were in contact through a diathermic boundary.  A boundary is Diathermic if a change of state is observed when two objects of different temperatures are brought into contact.

 


An adiabatic wall is a boundary where no change occurs between two objects with different temperature.  (insulating walls)

Fig. 1.3

 

The Zeroth Law of Thermodynamics: If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C is also in thermal equilibrium with A.

 

The Zeroth law justifies the concept of temperature and the use of a thermometer.  (e.g. mercury thermometer)


 

 


Early experiments with liquid thermometers arbitrarily divided the freezing point and boiling point of water into 100 units to give the Celcius temperature scale of 0 °C. for ice and 100 °C for boiling water  at 1 atm pressure.  Due to use of many liquids/materials in the thermometer, different expansions occurred resulting in different scales.

 

The pressure of a gas can be used to construct a perfect-gas temperature scale (virtually independent of gas). This is also identical to the thermodynamic temperature scale.

 

Temperature on the thermodynamic temperature scale is reported in kelvins, K.  

 

T/K = q/°C + 273.15 (definition of Celsius temp. in terms of kelvin temp.)

Gas Laws

 

Boyle 1661 showed    PV = constant   - Boyle’s Law

 

 

Fig. 1.5


 

 

 

 


 

 



Fig. 1.6

 

 

 

 


Molecular interpretation of Boyle’s Law:

 

As a sample is compressed to half its volume, twice as many molecules strike the walls in a given period of time than before compression.  Results in doubling average force on the walls.

 

Valid only at low pressures (limiting law).  

 

 

 

Next important addition came from Jacques Charles:

 

Volume increases linearly with temperature independent of the identity of the gas. 

 

Charles Law: V = constant  x T ( at constant pressure)

 



Fig. 1.7                                                  Fig. 1.8

 

 


Molecular explanation of Charles’s Law:

Raising temperature of a gas increases the average speed of its molecules.  The molecules collide with the wall more frequently and with greater impact, thus exerting greater pressure.

 

 

Final piece of data:  At a given pressure and temperature, the molar volume, Vm = V/n, volume per mole of molecules, of a gas is the same regardless of the identity of the gas.

 

V = constant x n (at constant T & P), the constant of proportionality is independent of the identity of the gas and only depends upon the number of molecules. 

 

Avogadro’s Principle: equal volumes of gas at same T&P have the same number of molecules.  

 

 

Combining the three contributions gives the equation of state for a perfect gas PV = nRT, where R is denoted as the gas constant.  This equation is approximate for any gas and becomes increasingly exact as the pressure of the gas goes to zero. 

 

 

 

Gas constant R is determined experimentally in the limit of pressure tending to zero.  R = 8.31451 J K-1 mol-1 =

8.20578 x 10-2 L atm K-1 mol-1 =

8.31451 x 10-2 L bar K-1 mol-1

 

 


 

 

 

 

 


Application of the Perfect Gas Equation

 

A commercial gas cylinder contains 75 L of Helium at 15 bar (gauge pressure).  Assuming ideal gas behavior for the isothermal expansion, how many 3.0 L balloons at a pressure of 1.1 bar can be filled by the gas in the cylinder?

 

 

Method - Summarize known and unknown data as follows:


 

 

 

 

 

 


 

 

 

 

V2 = V1 (P1/P2) = 75 L (16 bar/ 1.1 bar) = 1100 liters

 

 

We need to subtract the 75 L left in cylinder to give 1025 L

to fill balloons at 3 liters each.  Which finally gives

 

1025 L/ (3L / balloon) = ~341 balloons.

 

 

Mixtures of gases:

 

Dalton’s Law: The pressure exerted by a mixture of perfect gases is the sum of the partial pressures of the gases.  

The partial pressure of a perfect gas is the pressure it would exert if it occupied the container alone.

 

The total Pressure P = PA + PB + PC + ....   (sum of partial pressures)  with each substance J,  PJ = nJRT/V

 

 

Example of using Partial pressures:

 

A 1 gram sample of liquid water is added to 1  L of N2 gas at 1.01 bar and 25 0C.  To what volume must the system be adjusted in order for all the water to evaporate? Assume ideal gas behavior.  What will be the total pressure of the system?  The vapor pressure of water is 0.0316 bar @ 25 0 C. 

 

All the water will evaporate when the partial pressure of water in the container equals its vapor pressure. The volume of the system must be such that the pressure of water vapor is 0.0316 bar. 

 

 

 


 


The Mole fraction, xj , is the amount of J expressed as a fraction of the total amount of molecules, n, in the sample.

 

xJ = nJ / n  where n = nA + nB + .....

It follows that   xA + xB + ....  = 1

 

We define Partial Pressure pJ , of a gas J in a mixture ( perfect or real) as pJ = xJp , where p is total pressure. 

 

Also   pA +  pB + .....  = (xA + xB + .... ) p = p

 

 

 

Example 1:

The pressure of a gaseous mixture of NH3 and N2 decreased from 1.5 kPa to 1 kPa after all the NH3 was absorbed from the mixture.  Assume ideal gas behavior and find initial composition of the mixture.

 

 

Partial pressure of N2 is 1 kPa , PT = PNH + PN  

 P(NH3) = PT - P(N2) = 1.5 kPa - 1 kPa = 0.5 kPa

 

from Dalton’s Law :  x(N2) = P(N2)/PT = 1 kPa/1.5kPa = 0.67    

                                   x(NH3) =  1- x(N2) = 0.33