## The Gas Laws

### Table of contents

- Introduction to the Gas Laws
- Boyle’s Law
- Charles Law
- Gay-Lussac’s Law
- Avogadro’s Law
- Molar volume of the gas under different conditions

## Introduction to the Gas Laws

Development of gas laws started in the 18^{th}, when scientist understood the need to relate temperature, pressure and volume. So scientists came up with various laws. These are:

## Boyle’s Law

The first quantitative relationship between the volume and pressure of a gas was studied experimentally by Robert Boyle in 1662. The studies were made with air at room temperature. He used mercury and a simple U-tube of the type shown in Fig no. 1. The pressure was increased by putting more mercury into the open limb. The volume of the air enclosed in the space above mercury in the shorter limb was noted each time. Similar experiments were repeated with other gases.

The following generalization was observed which is known as **Boyle’s Law**:

**For constant temperature, the volume of a given mass of a gas is inversely proportional to its pressure.**

Mathematically, Boyle’s law may be stated as V ∝ 1/P

For a given mass of a gas at constant temperature.

or

that is, **PV = constant at constant temperature**

Where P and V represent the pressure and volume of the gas while k is a constant which depends upon the mass of the gas and temperature. The constant value obtained for PV is called **Boyle Constant**.

Thus, Boyle’s Law may also be stated as follows:

**At constant temperature, the product of volume and pressure of a given mass of a gas is constant**.** **

Thus, if P_{1} and V_{1} are the initial pressure and volume of a gas and keeping the temperature constant, if pressure is changed to P_{2}, then volume will change to V_{2} such that according to Boyle’s law

**P _{1}V_{1} = P_{2}V_{2} at constant temperature**

Thus if three variables are known the fourth can be easily calculated using above equation.

### Verification of Boyle’s Law

**(i) Experimental Verification of Boyle’s law.**

**Fig. No. 2 Verification of Boyle’ Law**

We observe that if the pressure is doubled at constant temperature, the volume is reduced to half. Similarly if pressure is made four times, the volume is reduced to 1/4 th and so on.

**(ii) Graphical Verification of Boyle’s law. **

According to Boyle’s law, P ∝ 1/v or PV = constant

Thus, the law can be verified by plotting

(i) P vs 1/v when a straight line passing through the origin is obtained fig no. 3 graph 1

or

(ii) PV vs P when a straight line parallel to the X-axis is obtained fig no. 3 graph 2

or

(iii) V vs P when branch of hyperbola in the first quadrant is obtained fig no. 3 graph 3

**Fig. No.3 Graphical Verification of Boyle’s Law**

In the equation, PV = constant, the value of the constant depends upon the amount of the gas taken (that is, no. of moles n) and the temperature (T). Hence, for a given amount of the gas taken, a different P-V curve is obtained at each temperature. Such a P-V curve at constant temperature is called an **Isotherm**. Three such isotherms are shown in Fig. No. 4a below:

Similarly, plots of P versus 1/V obtained at different temperatures are shown in Fig. No. 4b

### Significance of Boyle’s Law

The proof that the gases are compressible is given by Boyle’s law. Density of gas increases if it is compressed. Hence, it can be concluded that

**For a fixed mass of the gas, density is directly proportional to pressure at constant temperature**.

Mathematically, if m is the mass of the gas and V is its volume, then density, d = m/V

By Boyle’s law,

Substituting this value of V, we get

Thus, at constant temperature, for a fixed mass of the gas, d P.

Therefore with increase in height that is, altitudes, air is less dense as the atmospheric pressure is low. Making it hard to breathe because of dearth of oxygen. The person feels uneasiness, headache etc. This is called **Altitude Sickness**. Therefore the mountaineers carry oxygen cylinders with them.

## Charles Law

Studies on the effect of temperature on the volume of a gas at constant pressure were first carried out by the French scientist, Jacques Charles in 1787 and then extended by Joseph Gay Lussac in 1802. The following generalization was observed which is known as **Charles' Law**:

**At constant pressure, for every one degree centigrade rise or fall in temperature, volume of gas increases or decreases by the ratio of 1/273 of its volume at 0°C**.

Mathematically,

where V_{t}, is the volume of the gas at t°C and V_{0} is its volume at 0°C.

The volume of a certain mass of a gas at any temperature can be calculated by the application of the above relation. Thus:

This implies that a gas at -273°C will have zero or no volume, that is, it will cease to exist. Below this temperature, the volume will be negative which is meaningless. In fact, no one has ever traced the rate of contraction in volume of any gas upto -273°C at constant pressure. In actual practice, all gases liquefy before this temperature is reached.

We conclude graphically by plotting the volume of a certain mass of a gas (along y axis) against temperature (along x-axis) at constant pressure. We get a graph consisting of a straight line as shown in figure no. 5

**Fig. No. 5 Volume of gas as Function of Temperature**

If we extrapolate this straight line, it will meet the temperature axis at -273°C. Thus, at -273°C, a gas occupies zero or no volume. This temperature has never been realized in any laboratory of the world so far.

At constant pressure, the straight line V - t plots obtained for different amounts of the same gas or for same amount of the different gases are different but each of them intersects the temperature axis at -273°C corresponding to zero volume as shown in fig. no. 6 below. Each line in the plots shown in fig no. 6 is called an **Isobar** because it gives a plot of V vs T at constant pressure. Higher the pressure less is the slope of the line.

**Fig. No. 6 Volume vs Temperature for different gases at 1 atm**

### Absolute Zero and Absolute Scale of Temperature

**The temperature at which a gas is supposed to have zero volume at its lowest possible hypothetical or theoretical temperature of -273°C, this temperature is called Absolute zero**.

Experiments have shown that the absolute zero of temperature is -273.15°C. However for most of the purposes, the approximate value of -273°C is used.

Lord Kelvin has suggested a new scale of temperature starting with -273°C as its zero. This scale of temperature is known as **Kelvin Scale** or **Absolute Scale**. The size of the degree on the Kelvin scale is the same as that on the centigrade scale. On this scale, the freezing point of water that is, 0°C is 273 K (Kelvin). Thus, any temperature on the centigrade scale can be converted to that on the Kelvin scale by just adding 273 to its value on the centigrade scale. Hence, the two scales are connected by the relation:

Kelvin Temperature = Centigrade temperature + 273

Or

** T K = t°C + 273**

where T and t° are the temperatures on the Kelvin and the Centigrade scales respectively. This scale of temperature is called **Thermodynamic Scale of Temperature**.

The advantage of Kelvin scale lies in the fact that the volume of a gas and its temperature

According to Charles’ law in the form already defined,

Putting 273 + t = T, the corresponding temperature on the Kelvin scale, we get

As V_{0} and 273 are constant, hence V_{t} T or simply, V T or V = kT

The numerical value of the constant k depends upon the amount of the gas taken and the pressure. The above relation gives another definition of Charles’ law as follows:**Pressure remaining constant, the volume of a given mass of a gas is directly proportional to its temperature in degrees Kelvin**.

The relation V∝ T implies that

**V/T** **= Constant at constant pressure**.

Thus, if V_{1} is the volume of the gas at temperature T_{1} (in degrees Kelvin) and keeping the pressure constant, temperature is changed to T_{2}, then the volume will change to V_{2 }such that

**V _{1}/T_{1} = V_{2}/T_{2 }**

**at constant pressure**

Thus, a plot of V vs T at constant pressure will be a straight line passing through the origin.

**Fig. No.7 Volume vs Temperature at constant Pressure**

### Significance of Charles’ Law

Density of air decreases and it expands when it is heated. Thus, hot air is lighter than the atmospheric air. This fact is used in filling hot air in the balloon which rise up for meteorological observations.

## Gay-Lussac’s Law

**Gay-Lussac’s Law / Amonton’s Law (Variation of Pressure with Temperature at Constant Volume) **

The law which governs the relationship between pressure and temperature of a gas at constant volume is similar to that between volume and temperature at constant pressure. It states that -**“At constant volume for every 1°C rise or fall in temperature, the pressure of a given mass of a gas increases or decreases by the ratio of 1/273 of its pressure at 0°C.”**

Mathematically,

where P_{t} and P_{0} are the pressures of a certain amount of a gas at 1°C and 0°C respectively.

As P_{0} and 273 are constants, hence P_{t} T or simply P ∝ T that is, P = kT

The numerical value of the constant k depends upon the amount of the gas taken and the volume.

Thus the above law may also be defined as follows:

**Pressure of a given mass of a gas is directly proportional to its temperature in degrees Kelvin at constant volume**.

The above statement is referred to as Gay Lussac’s law or Amonton’s law.

Thus, at constant molar volume, the plot of pressure versus temperature (kelvin) will be straight line. Such a plot is called **Isochore** (as volume remains constant)

In 1703, G. Amonton constructed an air thermometer based on the principle that the pressure of a gas is a measure of the temperature of the gas.

The relation P ∝ T implies that

P/T = Constant at constant volume

which means that

P_{1}/T_{1} = P_{2}/T_{2} at constant volume

**Fig. No. 8 Pressure vs Temperature at Constant Volume**

## Avogadro’s Law

Avogadro’s law states that

**All gases having equal volumes under the same conditions of temperature and pressure contain equal number of molecules.**

As 1 mole of a gas contains Avogadro’s number of molecules (6.022 x 10^{23}), this means that one mole of each gas at the same temperature and pressure should have the same volume.

## Molar volume of the gas under different conditions

**(a) Standard temperature and pressure (STP) conditions**

**0°C** or **273.15 K** (freezing temperature of water) and **one atmospheric pressure**. Under these conditions, 1 mole of the gas occupies a volume of 22.413996 L = 22.4 L

However, if the standard conditions used are 0°C and 1 bar pressure, then as 1 bar < 1 atm

(1 bar = 0.987 atm), molar volume is slightly higher and equal to 22.71098 L mol^{-1} = 22.7 L mol^{-1}.

**(b) Standard ambient temperature and pressure (SATP) conditions **

**298.15 K** and **1 bar** (that is, exactly 10^{5} pascal) which is the atmospheric pressure at sea level. Under these conditions, the molar volume of an ideal gas is 24.789 L mol^{-1}.

Thus, according to Avogadro’s Law,

V ∝ n (number of moles of the gas) or V = kn

But n = m/M (m = mass, M = molar mass of the gas)

Therefore

Thus, we conclude that density of a gas is directly proportional to its molar mass.

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