JEE Chemistry States Of Matter Notes Set 03

Gas Laws and the Ideal Gas Equation

A gas has no fixed shape or volume and fills its container. Its state is described by pressure P, volume V, temperature T (always in kelvin) and amount n. Atmospheric pressure supports a mercury column of 760 mm at sea level: 1 atm = 760 mm Hg = 760 torr = 101325 Pa, and 1 bar = \( 10^5 \) Pa.
Boyle's law: at constant T and n, \( P_1V_1 = P_2V_2 \). Charles' law: at constant P and n, \( V_1/T_1 = V_2/T_2 \). Gay-Lussac's law: at constant V and n, \( P_1/T_1 = P_2/T_2 \). Avogadro's law: equal volumes of gases at the same T and P contain equal numbers of molecules. Combining them gives the ideal gas equation \( PV = nRT \), where \( R = 0.0821 \) L atm K\(^{-1}\) mol\(^{-1}\) = 8.314 J K\(^{-1}\) mol\(^{-1}\). One mole of an ideal gas occupies 22.4 L at STP (0 \(^\circ\)C, 1 atm).

Gas Mixtures and Dalton's Law

In a mixture of non-reacting gases each gas behaves as if it alone occupied the container. Dalton's law: \( P_{total} = p_1 + p_2 + p_3 + \dots \), with \( p_i = x_i P_{total} = n_iRT/V \), where \( x_i \) is the mole fraction. The average molar mass of a mixture is \( \sum x_iM_i \).
When gases are mixed or flasks are connected at constant temperature, the total of PV is conserved: \( P_{final} = \sum P_iV_i / V_{total} \). If two connected flasks are kept at different temperatures, the pressure is the same in both, so the moles are inversely proportional to the temperatures: \( n_1T_1 = n_2T_2 \) for equal volumes. A gas collected over water has pressure \( P_{dry} = P_{total} - \) aqueous tension.

Graham's Law of Diffusion and Effusion

At the same T and P the rate of diffusion (or effusion) of a gas is inversely proportional to the square root of its density or molar mass: \( r_1/r_2 = \sqrt{d_2/d_1} = \sqrt{M_2/M_1} \). The rate may be measured as a volume per second, moles per second or distance per second. For equal volumes the times are in the ratio \( t_1/t_2 = \sqrt{M_1/M_2} \). At unequal pressures, rate \( \propto P/\sqrt{M} \) at the same T. When two gases enter a tube from opposite ends, they meet at distances proportional to their rates. Lighter gases such as H\(_2\) and He leak and diffuse fastest.

Kinetic Theory of Gases

The kinetic theory assumes that a gas consists of tiny, widely separated molecules in constant random motion, that collisions are perfectly elastic, that there are no attractions or repulsions between molecules, and that the average kinetic energy is proportional to the absolute temperature. The pressure follows from \( PV = \frac13 mNu^2_{rms} \).
The three molecular speeds are \( u_{rms} = \sqrt{3RT/M} \), \( u_{av} = \sqrt{8RT/\pi M} \) and \( u_{mp} = \sqrt{2RT/M} \). Their ratio is \( u_{mp} : u_{av} : u_{rms} = 1 : 1.128 : 1.225 \). The average kinetic energy of a molecule is \( \frac32 kT \) and of one mole is \( \frac32 RT \). It depends only on temperature, not on the nature of the gas. Raising the temperature shifts the Maxwell speed distribution to the right, flattens the peak and keeps the area under the curve equal to one.

Eudiometry

Eudiometry is the analysis of gas volumes by explosion or absorption in a graduated tube. At the same T and P, volumes are in the same ratio as moles, so the balanced equation can be read directly in volumes. A hydrocarbon \( \ce{C_xH_y} \) burns as \( \ce{C_xH_y + (x + y/4)O2 -> xCO2 + (y/2)H2O} \). Cooling condenses steam, KOH absorbs CO\(_2\), alkaline pyrogallol absorbs O\(_2\) and turpentine oil absorbs O\(_3\). If a compound contains nitrogen it is usually converted to N\(_2\); S and P give SO\(_2\) and P\(_4\)O\(_{10}\). Ozone decomposes as \( \ce{2O3 -> 3O2} \), which increases the volume by half of the ozone volume.

Worked Examples

Example 1: At constant T, a gas at 1 atm in 2 L is compressed to 0.5 L. By Boyle's law \( P_2 = 1 \times 2/0.5 = 4 \) atm.
Example 2: The vapour density of a gas is 22. Its molar mass is 44 g/mol, and its rate of effusion relative to H\(_2\) is \( \sqrt{2/44} = 0.213 \).
Example 3: 20 mL of CH\(_4\) is burnt with excess O\(_2\). CO\(_2\) formed is 20 mL, and after steam is condensed the contraction is \( 20 + 40 - 20 = 40 \) mL.
Example 4: The rms speed of O\(_2\) at 300 K is \( \sqrt{3 \times 8.314 \times 300/0.032} = 484 \) m/s.