# Gases and gas law

Gas constant:

The gas constant R is not only of simple constant, but it is the amount of work done per mole per Kelvin. For n moles of a gas,

P V = n R T

Absolute zero of temperature:

The temperature, at which volume or pressure of any gas reduces to zero, is referred to as absolute zero temperature. Its value is equal to at 0 degrees Kelvin or -273 degrees Celsius or at -460 degrees Fahrenheit.

Charle’s pressure law:

Charle’s pressure law states that at constant volume, the pressure of the given mass of a gas increase or decreases by constant fraction of its pressure at 0oC for each degree rise of fall in temperature.

Let consider a gas at a [pressure of ${{\rm{p}}_{\rm{o}}}{\rm{\: at\: }}{0^{\rm{o}}}{\rm{C\: and\: p\: at\: }}{\theta ^{\rm{o}}}{\rm{C}}.{\rm{then\: }}$from above statement, we can write

${\rm{p}} = {{\rm{P}}_{\rm{o}}} + {\rm{\: }}\left( {1 + {\gamma _{\rm{v}}}\theta } \right)$

Where

${\gamma _{\rm{v}}}{\rm{\: }}$ Is constant called coefficient of expansion of gas at constant volume or pressure coefficient of gas.

Or,
${\rm{p}} = {{\rm{P}}_{\rm{o}}} + {{\rm{p}}_{\rm{o}}}{\gamma _{\rm{v}}}\theta$

Experimentally it is found that the value of γv is $\frac{1}{{273}}$

${\rm{p}} = {{\rm{P}}_{\rm{o}}} + {{\rm{p}}_{\rm{o}}}\frac{1}{{273}}\theta$

${\rm{p\: }}$=${\rm{\: }}{{\rm{p}}_{\rm{o}}}\frac{{\left\{ {273 + \theta } \right\}}}{{273}}$

${\rm{p}} = {{\rm{p}}_{\rm{o}}}\frac{{\rm{T}}}{{{{\rm{T}}_0}}}$

Where 273+θ = T is the absolute temperature at θ0 C and 273 = Tois absolute temperature at zero degree centigrade.

$\frac{{{{\rm{p}}^{\rm{o}}}}}{{{{\rm{T}}_0}}} = \frac{{\rm{p}}}{{\rm{T}}} = {\rm{constant}}$

Or, $\frac{{\rm{p}}}{{\rm{T}}} = {\rm{constant}}$

${\rm{P}} \propto {\rm{T}}$

This gives thecharle’s pressure law

The equation of state for an ideal gas:

Those gas which obeys ideal gas equation i.e PV = nRT is known as ideal gas. At high temperature and low pressure a gas behaves as an ideal gas.

Boyle’s law states that for a fixed amount of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional i.e

${\rm{p}} \propto \frac{1}{{\rm{v}}}$

Or PV = K. Graphical representation of law is as follow:

graph between P vs V, P vs 1/V, PV vs

V and PV vs P respectively

The equation states that product of pressure and volume is a constant for a given mass of confined gas as long as the temperature is constant. For comparing the same substance under two different sets of condition, the law can be usefully expressed as

${{\rm{P}}_1}{{\rm{V}}_1} = {{\rm{P}}_2}{{\rm{V}}_2}{\rm{\: }}$

The equation shows that, as volume increases, the pressure of the gas decreases in proportion. Similarly, as volume decreases, the pressure of the gas increases.

Charles's law states that if a given quantity of gas is held at a constant pressure, its volume is directly proportional to the absolute temperature. It can be represented as

${\rm{V}} \propto {\rm{T}}$

Or V =kT where k is a proportionality constant.

When temperature increases the volume of a certain mass of gas at constant pressure increases,

The graph given below state charle’slaw:

This law describes how a gas expands as the temperature increases; conversely, a decrease in temperature will lead to a decrease in volume. For comparing the same substance under two different sets of conditions, the law can be written as:

$\frac{{{{\rm{V}}_1}}}{{{{\rm{T}}_1}}} = \frac{{{{\rm{V}}_2}}}{{{{\rm{T}}_2}}}{\rm{\: \: or}}\frac{{{{\rm{V}}_2}}}{{{{\rm{V}}_1}}} = \frac{{{{\rm{T}}_2}}}{{{{\rm{T}}_1}}}{\rm{\: or\: }}{{\rm{V}}_1}{{\rm{T}}_2} = {{\rm{V}}_2}{{\rm{T}}_1}$

The equation shows that, as absolute temperature increases, the volume of the gas also increases in proportion.

Let the pressure, volume and temperature of a certain mass of a gas be denoted by P, V and T respectively,

Then by Boyle’s law, we have;

${\rm{P}} \propto \frac{1}{{\rm{v}}}$…… i)

According to Charles’s law we have

${\rm{v}} \propto {\rm{T}}$…. ii)

Further the volume of a gas is directly proportional to the number of moles according to Avogadro’s law which states that “equal volumes of all gases contain equal number of molecules under same condition of temperature and pressure.”

So,

${\rm{V}} \propto {\rm{n\: }}$….. iii)

From i) ii) and iii) we get

V = nRT /P…..iv)

Where R is proportionality constant and is defined as universal gas constant i.e. the constant used when 1gram mole of each gas is taken under consideration.

From equation iv)

We have PV= nRT Hence proved.

When gas obeys Boyle’s law then prove that pressure coefficient and volume coefficient are equal.

The gases do not obey Bayle’s strictly at all values of temperature and pressure. This obeyed by the gases at high temperature and low pressure but is not obeyed at temperature and high pressure.

Boyle’s law states that for a fixed amount of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional i.e

${\rm{p}} \propto \frac{1}{{\rm{v}}}$

Or PV = K. Graphical representation of law is as follow:

graph between P vs V, P vs 1/V, PV vs

V and PV vs P respectively

The equation states that product of pressure and volume is a constant for a given mass of confined gas as long as the temperature is constant. For comparing the same substance under two different sets of condition, the law can be usefully expressed as

${{\rm{P}}_1}{{\rm{V}}_1} = {{\rm{P}}_2}{{\rm{V}}_2}{\rm{\: }}$

The equation shows that, as volume increases, the pressure of the gas decreases in proportion. Similarly, as volume decreases, the pressure of the gas increases.

Air pressure in a car tire increases during driving:

Air pressure in a car tire increases during driving, temperature of air inside increases while its volume remains constant According to charle;s law

${\rm{V}} \propto {\rm{T}}$

At constant volume pressure P is directly proportional to time T therefore pressure inside tires increases.

At very low pressure and high temperature, the real gases behave like ideal gases:

At low temperature and high pressure, the molecular attraction between the molecules becomes appreciable. In addition, the volume of the gas molecules cannot be ignored in comparison to the volume of the gas. Due to these factors, the gas shows deviation from ideal gas behavior and hence it does not obey gas laws.

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