General Gas Equation

When one sample of gas is described under two different conditions, the ideal gas equation must be applied twice - to an initial condition and a final condition. This is: 

Standard Conditions for Comparing Gas Samples

Standard Temperature and Pressure (STP) 0.0°C (273.15 K) and 1 atm pressure. Thus the volume of 1 mol of an ideal gas is 24.79 L at SATP (Standard Ambient Temperature and Pressure which is 25 degrees Celcius and 1 bar of pressure) and 22.41 L at STP, approximately equivalent to the volume of three basketballs.

Mixtures of Perfect Gases

The perfect gas law assumes that all gases behave identically and that their behavior is independent of attractive and repulsive forces. If volume and temperature are held constant, the perfect gas equation can be rearranged to show that the pressure of a sample of gas is directly proportional to the number of moles of gas present : P = n x (RT/V). Let’s suppose we have a mixture of two perfect gases that are present in equal amounts. What is the total pressure of the mixture? The total pressure exerted by a mixture of gases is the sum of the partial pressures of component gases. This law was first discovered by John Dalton, the father of the atomic theory of matter. It is now known as Dalton’s law of partial pressures. 

Mole Fractions of Gas Mixtures

The composition of a gas mixture can be described by the mole fractions of the gases present. The mole fraction(X) of any component of a mixture is the ratio the number of moles of that component to the total moles of all species in a mixture(nt), Xa = moles of a /total moles = na/na+nb. 

So if we want to calculate the partial pressure we will get the partial pressure of a component/species is Xa = Pa / Ptot or Pa = Xa x Ptot.

Kinetic Theory of Gases

The kinetic molecular theory of gases is based on the following five postulates:

• A gas is composed of a large number of particles called molecules (whether mono-atomic or poly-atomic) that are in constant random motion.

• Because the distance between gas molecules is much greater than the size of the molecules, the volume of the molecules is negligible.

• Inter-molecular interactions, whether repulsive or attractive, are so weak that they are also negligible.

• Gas molecules collide with one another and with the walls of the container, but these collisions are perfectly elastic; that is, they do not change the average kinetic energy of the molecules.

• The average kinetic energy of the molecules of any gas depends on only the temperature, and at a given temperature, all gaseous molecules have exactly the same average kinetic energy.

For three dimensions of particle motion, we calculate the average translation kinetic energy of molecules of gas which is E = 3/2 kT. The molecules collides with the wall will create impulse  or momentum transfer ( 2mv) and the frequency of the collision itself can be expressed as the number of molecules collides over the area per second : f = v x (N/Vol).

Since the pressure happened of those molecules, P is the multiplication of impulse and collision frequency and the velocity of the molecules is root mean square of velocity of each directions, we will get PV = NkT, k is constant per molecule, if we use Avogadro 's number Na then R is the constant per molecule : PV = nRT.

( Note : really sorry for the formula or equation writing, because we still have difficulty to write the equation on web ).

The Relationships among Pressure, Volume, and Temperature

• Pressure versus Volume: At constant temperature, the kinetic energy of the molecules of a gas and hence the rms speed remain unchanged. If a given gas sample is allowed to occupy a larger volume, then the speed of the molecules does not change, but the density of the gas (number of particles per unit volume) decreases, and the average distance between the molecules increases. Hence the molecules must, on average, travel farther between collisions. They therefore collide with one another and with the walls of their containers less often, leading to a decrease in pressure. Conversely, increasing the pressure forces the molecules closer together and increases the density, until the collective impact of the collisions of the molecules with the container walls just balances the applied pressure.

• Volume versus Temperature: Raising the temperature of a gas increases the average kinetic energy and therefore the rms speed (and the average speed) of the gas molecules. Hence as the temperature increases, the molecules collide with the walls of their containers more frequently and with greater force. This increases the pressure, unless the volume increases to reduce the pressure, as we have just seen. Thus an increase in temperature must be offset by an increase in volume for the net impact (pressure) of the gas molecules on the container walls to remain unchanged.

• Pressure of Gas Mixtures: Postulate 3 of the kinetic molecular theory of gases states that gas molecules exert no attractive or repulsive forces on one another. If the gaseous molecules do not interact, then the presence of one gas in a gas mixture will have no effect on the pressure exerted by another, and Dalton’s law of partial pressures holds.

Real Gases

The postulates of the kinetic molecular theory of gases ignore both the volume occupied by the molecules of a gas and all interactions between molecules, whether attractive or repulsive. In reality, however, all gases have nonzero molecular volumes. Furthermore, the molecules of real gases interact with one another in ways that depend on the structure of the molecules and therefore differ for each gaseous substance. What is important to note here, is that an ideal gas can exist only as a gas. It is not possible for an ideal gas to condense into some kind of “ideal liquid”.

The Dutch physicist Johannes van der Waals (1837–1923; Nobel Prize in Physics, 1910) modified the ideal gas law to describe the behavior of real gases by explicitly including the effects of molecular size and intermolecular forces. In his description of gas behavior, the so-called van der Waals Equation :