## A few of JEE advanced important topics from Ideal Gas Law

Dalton’s Law of Partial Pressures

Mixtures of gases are often used in experiments. The total gas pressure is related to the partial pressures, that is, the pressures of the individual components of the mixture. Dalton formulated a law, which is now known as Dalton’s law of partial pressures, the total pressure of a mixture of gases equals the sum of the pressures that each gas exerts alone.

Consider the case in which two gases A and B are in a volume V vessel. The pressure exerted by gas A, according to the ideal gas equation, is:

$P_{A} = \frac{n_{A}RT}{V}$

Where nA is the number of moles of A present. In the same way, the pressure exerted by gas B is:

$P_{B} = \frac{n_{B}RT}{V}$

In a mixture of gases A and B, the total pressure Pt is the result of the collisions of both types of molecules A and B, with the walls of the container. Therefore, according to Dalton’s law, Pt = PA + PB

$P = \frac{n_{A}RT}{V} + \frac{n_{B}RT}{V} = \frac{RT}{V} \left ( n_{A} + n _{B}\right )$

Where n, is the total number of moles present. Thus, for a mixture of gases, the PT depends only on the total number of moles of gas present, not on the nature of the gas molecules.

To realize how each partial pressure is related to the total pressure:

$\frac{P_{A}}{P_{B}} = \frac{\frac{n_{A}RT}{V}}{(n_{A}+n_{B})\frac{RT}{V}} = \frac{n_{A}}{(n_{A})+n_{B}} = X_{A}$

Where XA is called the mole fraction of gas A, expresses the ratio of the number of moles of a component to the number of moles of all the components present. The partial pressure of A can be expressed as:

PA = XAPT

In the same way: PB = XBPT

If a system has more than two gases, the partial pressure of component i is related to the total pressure by: Pi = Xi PT

Vapor pressure

The vapor pressure of a substance depends only on the temperature and not on the volume; that is, a container that contains liquid and vapor in equilibrium at a fixed temperature, the pressure is independent of the relative amounts of liquid and vapor present.

As the temperature of the liquid increases, a greater number of molecules have sufficient kinetic energy to escape from the liquid phase, the flow of escaping molecules is greater and therefore, at equilibrium, the saturated vapor pressure will be even greater.

$\mu _{\alpha }(T,P) = \mu _{\beta}(T,P)$

The boiling temperature is that for which, the vapor pressure is equal to the external pressure. The vapor pressure of the water is equal to one atmosphere at the temperature of 100ºC.

Most materials expand when their temperature increases, so that a certain mass of material increases its volume decreasing its density.

Water presents a quite complex behavior. Ice (solid phase) is less dense than water (liquid phase), which causes ice to float on water.

The density of water at 0ºC is 999.8 kg / m3, reaches a maximum at a temperature close to 4ºC and then decreases with the increase in temperature (normal behavior). The coefficient of expansion of the water is therefore negative in the range between 0ºC and 4ºC, and positive from that temperature.

Saturated Steam Pressure

If the process of evaporation takes place in a closed container, there will come a time when there are as many molecules returning to the liquid state as those that escape the gas state. At this point, it is said that the vapor is saturated, and the pressure of that vapor (usually expressed in mmHg), is called saturated vapor pressure.

Since the molecular kinetic energy is higher at higher temperature, more molecules can escape from the surface and consequently the saturated vapor pressure is higher. If the liquid is open to air, then the vapor pressure is estimated as a partial pressure, along with the other constituents of the air. The temperature at which the vapor pressure is equal to the atmospheric pressure is called the boiling point temperature.

The Clausius-Clapeyron equation

The evaporation of water is an example of a phase change from liquid to vapor. The chemical potentials of the phases α (liquid) and β (vapor) are functions of temperature T and pressure P and have the same value

μα (T, P) = μβ (T, P)

From this equality and using thermodynamic relations, the Clapeyron equation is obtained.

Assuming that the vapor phase is an ideal gas and that the molar volume of the liquid is negligible compared to the molar volume of gas, we reach the so-called Clausius-Clapeyron equation that gives us the vapor pressure of water Pv as a function of the temperature T, assuming further, that the enthalpy L of vaporization is independent of temperature (at least in a certain interval)

ln ⁡P v = -L R (1 T) + C

where C is a constant

Diffusion of gases

Diffusion, that is, the gradual mixing of the molecules of one gas with molecules of another gas, by virtue of their kinetic properties, constitutes a direct demonstration of random movement. The diffusion always comes from a region of greater concentration to another less concentrated one. Although the molecular speeds are very large, the diffusion process takes a relatively long time to complete. Therefore, the diffusion of gases always happens gradually. In addition, since the root of the average square velocity of a light gas is greater than that of a heavier gas, a light gas will diffuse through a certain space faster than a heavy gas.

Gas diffusion

It is the phenomenon by which the molecules of one gas are evenly distributed to the other gas. It is also established as the ability of gaseous molecules to pass through small openings, such as porous walls, ceramic or porcelain that is not glazed.

Law of Gas Dissemination

It was established by Thomas Graham; who manifests the following:

“Under the same pressure and temperature conditions, the diffusion rates of two gases are inversely proportional to the square roots of their molecular masses.”

Analysis:

Let’s call M1 the mass of the molecules of one species and M2 the mass of the molecules of another species. Then, the average kinetic energies of the molecules of each gas are given by the expressions:

0.5 M1 v12 = k . T . 3/2

and

0.5 M2 v22 = k . T . 3/2

because the temperature is the same. Dividing member to member we have that:

M1 / M2 = v12 / v22

that is, the quotient of the square root of the square of the average velocity for both species is inversely proportional to the mass of that species. In formula:

v1 / v2 = (M1 / M2)0.5

Since the mass is proportional to the density and the quotient of the left limb is a measure of the speed with which the molecules of one species move with respect to the other and this is precisely the mechanism underlying the diffusion, this equation is the mathematical expression of Graham’s law.

If there are still some doubts you can always refer to this book.

This site uses Akismet to reduce spam. Learn how your comment data is processed.