Subject: Chemistry
Application of Charle's law
a) Charle's law is used for making hot air balloon. From Charle's law, when the temperature increases, the gas becomes lighter and easily rises up by displacing atmospheric gas downwardly. These hot air balloons are used as transport and in various experimental observation.
Relation between temperature and pressure of a gas
It staters that " at constant volume, the pressure exerted by the gas is directly proportional to its absolute temperature'.
Mathematically, P∝ T at constant volume
or, P = kT
or, \( \frac {P}{T}\) = k
\( \frac {P_1}{P_2}\) =\( \frac {T_1}{T_2}\)
This law is known as Gay - Lussace's law
Avogadro's law
The volume of the gas not only depends on the pressure and temperature but also in its amount.
It states that ' at constant temperature and pressure, the volume of a gas is directly proportional to the total amount of the gas '.
mathematically,
V∝n at constant T and P
or, V = n . constant
or, \( \frac {V}{n}\) = constant
The constant is same for all gases. So, Avogadro's law can also be defined as " Under the similar condition of temperature and pressure, the equal volume of gas contains the same number of molecules (no. of mole)". At NTP, 1 mole of any gas contains 8.023 x 10^{23} number of molecules and occupies 22.4 liters. This number is called Avogadro's number and this volume is called molar volume.
Combined gas equation
From Boyle's law
V∝ \( \frac {1}{P}\) at constant temperature
From Charle's law,
V∝ T at constant pressure
From above equations,
V∝\( \frac{T}{P}\) , under similar condition of temperature and pressure
PV∝ T
or, PV = kT
or, \( \frac {PV}{T}\) = k -(i)
Hence,
\( \frac{P_1V_1}{T_1}\) = \( \frac{P_2V_2}{T_2}\) ------(ii)
Ideal gas equation
The mathematical equation which is obeyed by gases at ideal condition is known as the ideal gas equation.
Derivation
Boyle’s law: V ∝ \( \frac {1}{P}\) (at constant temperature) --------(i)
Charles law: V ∝ T (at constant pressure) -------------(ii)
Avogadro’s law: Equal volume of all the gases contains an equal number of molecules/ moles under the condition of same temperature and pressure.
V ∝ n (at constant temperature and pressure) -----------(iii)
Combining equation (i), (ii) and
V ∝ \( \frac{nT}{P}\)
PV∝ nT
or, PV = nRT -(i)
where proportionaly constant 'R' is called gas constant and the equation (i) is called ideal gas equation.
Universal gas constant R:
The gas constant 'R' present in the ideal gas equation is called the universal gas constant. It is because the value of R is same for all types of ideal gas.
Ideal gas equation: PV = nRT
Or, R = \( \frac {PV}{nT}\)
= \( \frac{Force}{Area}\) x \( \frac{Volume}{No\:of\:mole\:× Temperature\:in\:Kelvin}\)
Volume = Area x Length
Therefore,
R == \( \frac{Force\:Length}{No\:of\:mole\:× Temperature\:in\:Kelvin}\)
Hence, the gas constant 'R' is defined as the amount of work done required for 1 mole of a gas to rise its temperature by 1K
Value of R in liter x atm
Pressure = 1atm
Volume = 22.4 lit
Number of mole (n) = 1
Temperature (T) = 273 K
PV = nRT
Or, R = \( \frac{PV}{nT}\)
or, R = \( \frac { 1× 22.4} { 273× 1}\)
Or, R = 0.0821l atm K^{-1} mol^{-1}.
Value of R in energy unit ( Joule and Calories)
Pressure = 1.01 x 10^{5 }Pa
Volume = 22.4 lit = 22.4 x 10^{-3} m^{3}
Number of mole (n) = 1
Temperature (T) = 273 K
PV = nRT
Or, R = \( \frac{PV}{nT}\)
or, R = \( \frac { 1.01 ×10^5 × 22.4 × 10^{-3}} { 273× 1}\)
Or, R = 8.31JK^{-1} mol^{-1}.
Again, 1 calorie = 4.2 Joules
Then, R = \( \frac{8.31}{4.2}\) cal K^{-1} mol^{-1}. = 1,987 cal K^{-1} mol^{-1}
Relation between pressure, density and molar mass from ideal gas equation
From ideal gas equation, PV = nRT -------(i)
where, P = Pressure
V = Volume
n = Number of moles
R = Universal Gas constant
T= Temperature
Since, Volume(V) = \(\frac{Mass}{Density}\) nad Number of mole = \( \frac{Given \:mass}{Molar\:Mass}\)
P.\( \frac {m}{d}\) = \( \frac{m}{M}\) RT
or, \( \frac{P}{d}\) = \( \frac{RT}{M}\)
or, PM = dRT
where M = molar mass of the gas
Relation between density, pressure and temperature from combined gas equation
From combined gas equation,
\( \frac{P_1V_1}{T_1}\) = \( \frac{P_2V_2}{T_2}\)
Since volume (V) = \( \frac{Mass(M)}{Density(d)}\)
\( \frac{P_1m_1}{d_1T_1}\) =\( \frac{P_2m_2}{d_2T_2}\)
For any particular gas, m_{1} = m_{2}
or, \( \frac{T_1d_1}{P_1}\) = \( \frac{T_2d_2}{P_2}\)
Dalton’s law of partial pressure
It was introduced by John Dalton in 1807. It is the law to calculate the pressure of the gaseous mixture from the partial pressure of the component gases.
Dalton’s law of partial pressure states that “the total pressure of the mixture of gases is equal to the sum of the partial pressure of the component gases at constant temperature and pressure.
Let P_{T }be the total pressure and P_{1}, P_{2} and P_{3} be the partial pressure of the gases A, B, and C respectively.
Then,
P_{T} = P_{1} + P_{2} + P_{3}
Mathematical deduction of Dalton's law
Let n_{1}, n_{2}, n_{3}be the number of moles of three non-reacting gases enclosed in a vessel at pressure P and volume V.
Then, from ideal gas equation
PV = nRT
or, PV =(n_{1}+n_{2}+n_{3}) RT
or, P = \(\frac{n_1}{V}\) RT +\(\frac{n_2}{V}\) RT +\(\frac{n_3}{V}\) RT
or, P= P_{1} + P_{2} + P_{3} [At constant temperature]
Dalton's law in terms of mole fraction
P= P_{1} + P_{2} + P_{3} [At constant temperature]
P_{1} =\(\frac{n_1}{V}\) RT ----(i)
P_{2} =\(\frac{n_2}{V}\) RT ----(ii)
P_{3} =\(\frac{n_1}{V}\) RT ----(i)
From ideal gas equation
P =\(\frac{n}{V}\) RT ----(iv)
Dividing (iv) by (i)
\( \frac{P_1}{P}\) = \( \frac{\frac{n_1}{V}RT}{\frac{n}{V}RT}\)
or,\( \frac{P_1}{P}\) =\( \frac{n_1}{n}\)
Hence, P_{1} =\( \frac{n_1}{n}\) P --(v)
P_{1} = X_{1}.P where, X_{1} = mole fraction of gas
P_{2} = X_{2}.P
P_{3} = X_{3}.P
In general,
P_{i} = X_{i}P --------(vi)
The mole fraction of the gas can be defined as the ratio of a number of mole of the individual component of gas to the total number of moles.
Applications of Dalton's law 's laws
a) Dalton's law of partial pressure is used to calculate the pressure exerted by pure (dry) gas which is collected over water. The pressure exerted by water is called water vapour pressure. It is calculated as
P_{dry} = P_{moist} - P_{water vapor}
The pressure exerted by water vapor at a particular temperature is called aqueous tension and is denoted by f. It is known for any temperature
P_{dry} = P_{moist} - Aqueous tnesion
Graham's law of diffusion
The property of a gas by virtue of which they intermix with each other is known as diffusion of gas.
The diffusion of gas is explained by Graham's law of diffusion. It states "at constant temperature and pressure, the rate of diffusion of different gases are inversely proportional to square root of their density"
Mathematically,
r∝ \( \frac{1}{\sqrt d}\)
Let r_{1} and r_{2} be the rate of diffusion of two gases having density d_{1} and d_{2}respectively. Then from Graham's law,
r_{1}∝ \( \frac{1}{\sqrt d_1}\) --(i)
r_{2}∝ \( \frac{1}{\sqrt d_2}\) --(ii)
KINETIC THEORY OF GASES
Postulates
i) The molecules of a gas do not exert any attractive force with each other, that means the intermolecular force of attraction is almost negligible.
ii) The pressure exerted by the gas molecule is due to the collision of gas molecules on the wall of the container in which it is contained.
iii) The average kinetic energy of a gas molecule is directly proportional to the absolute temperature i.e. molecules have the same level of energy at fixed temperature.
iv) Different molecules move with different motion/speed. Further, the speed of molecules goes on increasing due to the collision. In spite of this fact, the distribution of molecular speed remains unchanged at fixed temperature This distribution of molecular speed is known as Maxwell - Boltzmann distribution
Qualitative explanation of Boyle's law and Charle's law
i) Boyle's law:From kinetic theory of gas, the pressure exerted by a gas is due to the collision of gas molecules on the wall of the container. For a fixed mass of gas, the average kinetic energy is also fixed which also fixes the average velocities.
For a fixe mass of gas, a total number of molecules will also be fixed. When the volume is decreased, a total number of molecules per unit volume will increase. This increased number of molecules also increases the collision of gas molecules increases and ultimately increases the pressure of the gas. On the other hand, if the volume is increased, the number of molecules per unit volume decreases. The molecules are far apart from each other which causes striking of molecules on the wall to decrease. Hence, the pressure deceases on increasing volume at the constant temperature which is Boyle's law.
ii) Charle's law
If a fixed mass of gas is heated in a container, or a fixed volume, its pressure increases. If the volume is not fixed ad pressure is kept constant, then, on heating the container, the gas expands and expansion continues till the pressure decreases to its original value. Hence, the result of the increase in temperature is increased in volume of the container at constant pressure which is Charle's law.
Ideal gas and Real Gas
A gas which obeys ideal gas equation PV = nRT at all conditions of temperature and pressure is called ideally, has. But, all known gas are found to obey ideal behavior at only low pressure and high temperature. The real gas shows the marked deviation from ideal behavior at high pressure and low temperature.
Deviation of real gas from ideal behaviour
The real gas deviates from ideal behaviour which can be explained by the following equation,
PV = nRT (for ideal gas)
and PV = z(nRT) (For real gas)
where 'z' measures the degree of deviation of real gas from ideal behavior.
When,
i) z = 1, PV = nRT , the real gas shows ideal behaviour under all conditions of temperature and pressure
ii) When z < 1, real gas shows negative deviation i.e. the gases are more compressible than the ideal gas.
iii) When z>1, the real gas shows positive deviation i.e. the gases are less compressible than ideal gas
The factor 'z' is called compressibly factor which measures the degree of deviation from ideal behaviour.
For a real gas, the effect of deviation can be explained in terms of temperature and pressure
i) Effect of pressure on deviation
The effect of pressure can be explained by ' z VS P' graph of different gases at a fixed temperature
ii) Effect of temperature on deviation
The effect of temperature can be explained by plotting 'z vs P' graph of particular gas at the different temperature.
The temperature at which real gas shows ideal behaviour over the wide range of pressure is called 'Boyle's temperature'.
i) When T < T_{B} , z < 1 at lower pressure and greater than 1 at higher pressure.
ii) When T > T_{B}, z > 1 at all pressure
Hydrogen and Helium gas always show positive deviation from ideal behaviour because the Boyle's temperature for these gases is quite low ( -165^{0}C for hydrogen and 240^{0}C for helium)
Cause of deviation
At low pressure and high temperature, gas shows ideal behaviour but at the reverse condition, (high pressure and low temperature) , real gas deviate from ideal behaviour . This fact may be attributed by following two faulty assumptions made in kinetic theory of gas:
i) The total volume occupied by a gas molecule is negligible as compared to the total volume of the gas
ii) There is no intermolecular force of attraction between gas molecules
Above two assumptions are valid at low pressure and high temperature because, at this conditions, gas molecules are far apart from one another. But at low temperature and high pressure, gas molecules come closer to another and its volume cannot be neglected. Similarly, when gas molecules come closer, they feel the same kind of operative force
To account this fact, Van Der Waal modified ideal gas equation as
( P + \( \frac{an^2}{V^2}) \) ( V - nb) = nRT
where 'an' and 'b' are Van Der Waal's constants
and this equation is called van Der Waal's equation
Equal volume of all the gases contains an equal number of molecules/ moles under the condition of same temperature and pressure.
It is the law to calculate the pressure of the gaseous mixture from the partial pressure of the component gases.
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