# Ideal Gas Equation

*Ideal Gas Equation –*

- A gas that follows Boyle’s law, Charles’ law and Avogadro law strictly is called an ideal gas

*Value of R –*

- At STP , Value of R for one mole of ideal gas is R= 8.314 J/K.mol

*Characteristics of Ideal gas –*

- It is hypothetical
- It obeys Boyle’s law, Charles Law and Avogadro’s Law at all conditions of temperature and pressure .
- Attractive forces among the molecules do not exist therefore an ideal gas cannot be converted into liquid or solid.

*Combined gas law*

- It is obtained by combining the Boyle’s law and Charles law .
- PV/T = k

**Density and Molar Mass of a Gaseous Substance**

__$M=\frac{dRT}{p}$ M=dRTp __

- Where d is density , R is gas constant , T is temperature , p is pressure and M is molar mass .

*Dalton’s Law of Partial Pressures*

- The law was formulated by John Dalton in 1801. It states that the total pressure exerted by the mixture of non-reactive gases is equal to the sum of the partial pressures of individual gases
- P
_{total}is the total pressure of mixture - p
_{1}, p_{2}, p_{3}is partial pressure of gases 1,2,3 …..

** ****Aqueous Tension**

Pressure of non reacting gases are generally collected over water and therefore are moist. Pressure of dry gas can be calculated by substracting vapour pressure of water from total pressure of moist gas.

P_{2Dry gas} = P_{Total} – Aqueous Tension

**Relation between Mole Fraction and Partial Pressure**

The relation between mole fraction and partial pressure can be explained through the following equations:

If three gases at a temperature T enclosed in volume V exert pressures P_{1}, P_{2} and P_{3}, then

From the ideal gas equation, we have

P_{1} = n_{1}RT/V , P_{2} = n_{2}RT/V and P_{3} = n_{3}RT/V

Where, n_{1},n_{2} and n3 are the number of moles of these gases.

The total pressure from Dalton’s Law of Partial pressures will be,

P_{total} = P_{1} + P_{2} + P_{3}

P_{total }= total pressure exerted by the mixture of gases.

P_{1}, P_{2}, P_{3}are partial pressures of individual gases.

On applying the ideal gas equation,

P_{1} = n_{1}RT/V , P_{2} = n_{2}RT/V and P_{3} = n_{3}RT/V

P_{Total} = P_{1} + P_{2} + P_{3}

P_{Total} = n_{1}RT/V + n_{2}RT/V + n_{3}RT/V

P_{Total} = (n_{1 }+ n_{2} + n_{3})RT/V

P_{1}/P_{Total} = n_{1}/(n_{1} + n_{2} + n_{3})

P_{1}/P_{Total} = n_{1}/n

n = n_{1} + n_{2} + n_{3}

n_{1}/n = x_{1}

P_{1}/P_{Total} = x_{1}

P_{1} = x_{1} P_{Total}

Similarly,

P_{2} = x_{2 }P_{Total}

P_{3} = x_{3} P_{Total}

Similarly for i^{th} gas

P_{i }= x_{i} P_{Total}

Here, P_{i} = partial pressure of i^{th} gas

x_{i} = mole fraction of i^{th} gas

The given equation can be used to determine the pressure exerted by individual gases in a mixture if the total pressure is known.

### Kinetic Energy and Molecular Speeds

Molecules of gases remain in continuous motion. While moving they collide with each other and with the walls of the container. This results in change of their speed and redistribution of energy. So the speed and energy of all the molecules of the gas at any instant are not the same. Thus, we can obtain only average value of speed of molecules. If there are n number of molecules in a sample and their individual speeds are u1, u2, …….un, then average speed of molecules uav can be calculated as follows:

Maxwell and Boltzmann have shown that actual distribution of molecular speeds depends on temperature and molecular mass of a gas. Maxwell derived a formula for calculating the number of molecules possessing a particular speed. Fig. shows schematic plot of number of molecules vs. molecular speed at two different temperatures T^{1} and T^{2} (T^{2} is higher than T^{1}). The distribution of speeds shown in the plot is called Maxwell-Boltzmann distribution

of speeds.

**Fig. :**** Maxwell-Boltzmann distribution of speeds **

The graph shows that number of molecules possessing very high and very low speed is very small. The maximum in the curve represents speed possessed by maximum number of molecules. This speed is called **most probable speed, **ump. This is very close to the average speed of the molecules. On increasing the temperature most probable speed increases. Also, speed distribution curve broadens at higher temperature. Broadening of the curve shows that number of molecules moving at higher speed increases. Speed distribution also depends upon mass of molecules. At the same temperature, gas molecules with heavier mass have slower speed than lighter gas molecules. For example, at the same temperature lighter nitrogen molecules move faster than heavier chlorine molecules. Hence, at any given temperature, nitrogen molecules have higher value of most probable speed than the chlorine molecules. Look at the molecular speed distribution curve of chlorine and nitrogen given in Fig. Though at a particular temperature the individual speed of molecules keeps changing, the distribution of speeds remains same.

* *

*Fig. : Distribution of molecular speeds for chlorine and nitrogen at 300 K *

We know that kinetic energy of a particle is given by the expression:

Therefore, if we want to know average translational kinetic energy, $\frac{1}{2}mu^2$12 `m``u`^{2} , for the movement of a gas particle in a straight line, we require the value of mean of square of speeds, $u^2$`u`^{2} , of all molecules. This is represented as follows:

The mean square speed is the direct measure of the average kinetic energy of gas molecules. If we take the square root of the mean of the square of speeds then we get a value of speed which is different from most probable speed and average speed. This speed is called **root mean square speed **and is given by the expression as follows:

$\mu rms$`μ``r``m``s`

Root mean square speed, average speed and the most probable speed have following relationship:

urms > uav > ump

The ratio between the three speeds is given below :

ump: uav : urms : : 1 : 1.128 : 1.224