Key Applications of the Ideal Gas Equation in Chemistry
Ideal gas Law
Ideal gas law is based on behavior of ideal gas. It is an approximation of the behavior of many real gases under many conditions. It is actually a combination of Boyle’s law, Charles’ Law and Avogadro’s law.
Charles’ Law and Avogadro’s law.
To understand the ideal gas law first you need to know Boyle’s law, Charles law, Avogadro’s law and Gay-Lussac’s Law.
Boyle’s law – Boyle’s law states that the absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies if temperature and amount of gas remain unchanged within a closed system.
Mathematical expression of Boyle’s law - P1V
where P is the pressure of the gas and V is the volume.
Charle’s Law – Charle’s Law states that when the pressure on a sample of a dry gas is held constant, the kelvi the voln temperature and volume will be in direct proportion.
Mathematical equation can be written as follows – V T,
where V is the volume of the gas and T is the temperature of the gas.
Gay-Lussac’s Law – Gay-Lussac’s law states that the pressure of a given mass of gas varies directly with the absolute temperature of the gas, when the volume is kept constant.
Mathematically it can be expressed as follows – P T ,
where P is pressure of the gas and T is temperature of the gas.
Avogadro’s law- Avogadro’s law states that at the same temperature and pressure, equal volumes of gases contain equal numbers of moles.
Mathematically it can be expressed as follows – V n ,
where V is volume of gas and n is number of moles of the gas
Now, let’s understand ideal gas law. Ideal gas law is expressed by the general gas equation which is a thermodynamics equation relating state variables such as pressure, volume and temperature with ideal gas.
Ideal gas law can be easily expressed by – PV = nRT
Ideal Gas Equation
Ideal gas law is expressed by an ideal gas equation.
The ideal gas equation is written as – PV = nRT
Where P = pressure of the gas
V = volume of the gas
n= number of moles of the ideal gas
R = Gas constant or ideal gas constant
T = temperature
Molar form of the ideal gas can be written as follows –
‘n’ number of moles of the gas is equal to the total mass of the gas divided by its molar mass. So, we can write n= m/M
Now let’s put the value of n in the above gas equation -PV = mMRT
Where m = total mass of the gas in kg
M = Molar mass (in kilograms per mole)
Derivation of Ideal Gas equation
If V = volume of the gas, P = pressure on gas and T = temperature then –
According to Boyle’s Law-
V 1P at constant T or temperature………………………………………. (I)
According to Charle’s Law-
V T at constant P or pressure……………………………………………… (II)
According to Avogadro’s Law-
V n at constant T and P………………………………………………………..(III)
Where n = number of moles of the gas
From equations (I), (II) and (III) we can write –
V ∝ 1P ×T ×n
We can write the above equation as –
V = R1P ×T×n , where R is the Universal Gas Constant and its value is 8.314 Jmol-1K-1
V = RTnP
After rearranging the above equation –
PV = nRT
Universal Gas Constant
Universal gas constant is also known as gas constant or ideal gas constant. It is denoted by ‘R’. it is an experimentally derived number that is used in ideal gas equations and other equations.
Ideal gas constant and its experimental values in different units are given below –
Applications of Ideal Gas Law
It is largely used in thermodynamics.
It can be used in stoichiometry problems.
It can be used to determine densities of gases.
Ideal gas law is used in the working mechanics of airbags which are used in vehicles.
Using ideal gas equations, we are able to use coolant gases in refrigerators, air conditioners etc.
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FAQs on Step-by-Step Derivation of the Ideal Gas Equation
1. How is the ideal gas equation derived from the fundamental gas laws?
The ideal gas equation, PV = nRT, is a composite formula derived by combining three fundamental gas laws that describe the relationships between pressure (P), volume (V), temperature (T), and the amount of gas (n).
- Boyle's Law: States that for a fixed amount of gas at constant temperature, the volume is inversely proportional to the pressure (V ∝ 1/P).
- Charles's Law: States that for a fixed amount of gas at constant pressure, the volume is directly proportional to the absolute temperature (V ∝ T).
- Avogadro's Law: States that at constant temperature and pressure, the volume of a gas is directly proportional to the number of moles (V ∝ n).
By combining these three proportionalities, we get: V ∝ (1/P) × T × n. This can be written as V = R(nT/P), where R is the proportionality constant. Rearranging this gives the final form of the ideal gas equation: PV = nRT.
2. What do the terms in the ideal gas equation, PV = nRT, represent?
Each variable in the ideal gas equation represents a specific physical property of an ideal gas:
- P stands for the pressure of the gas, typically measured in Pascals (Pa) or atmospheres (atm).
- V represents the volume occupied by the gas, measured in cubic meters (m³) or liters (L).
- n is the number of moles of the gas, which is a measure of the amount of substance.
- R is the universal gas constant, a constant of proportionality. Its value depends on the units used for other variables, but it is commonly 8.314 J·K⁻¹·mol⁻¹ or 0.0821 L·atm·K⁻¹·mol⁻¹.
- T denotes the absolute temperature of the gas, which must be measured in Kelvin (K).
3. What are the key assumptions made about an ideal gas for its derivation to be valid?
The derivation of the ideal gas equation is based on the kinetic theory of gases, which makes two critical assumptions about the behavior of gas particles:
- Negligible Molecular Volume: The volume occupied by the gas molecules themselves is considered to be infinitesimally small and negligible compared to the total volume of the container.
- No Intermolecular Forces: It is assumed that there are no attractive or repulsive forces between the gas molecules. The only interactions are perfectly elastic collisions with each other and the container walls.
Real gases deviate from this behavior, especially at high pressures and low temperatures.
4. How does the ideal gas equation differ from the real gas equation (van der Waals equation)?
The primary difference lies in the assumptions about gas behavior. The ideal gas equation (PV=nRT) assumes gas particles have no volume and no intermolecular forces. In contrast, the real gas equation, specifically the van der Waals equation, corrects for these factors:
- It introduces a correction term 'b' to account for the finite volume of gas molecules, effectively reducing the available volume to (V - nb).
- It adds a correction term 'a' to account for the intermolecular forces of attraction, which reduces the pressure exerted by the gas on the container walls.
Therefore, the ideal gas law is a simplification, while the van der Waals equation provides a more accurate description of real gases under various conditions.
5. What are some practical applications of the ideal gas equation?
The ideal gas equation is a cornerstone of thermodynamics and chemistry with numerous real-world applications. For instance, it is used to:
- Perform stoichiometric calculations involving gaseous reactants and products in chemical reactions.
- Determine the densities or molar masses of unknown gases.
- Understand the working mechanics of vehicle airbags, which inflate rapidly due to a chemical reaction producing a specific volume of gas.
- Design and operate systems involving gases, such as refrigerators and air conditioners that rely on the pressure-volume-temperature relationships of coolant gases.
6. How can the ideal gas equation be derived from the Kinetic Theory of Gases?
Deriving the ideal gas equation from the Kinetic Theory of Gases provides a deeper, microscopic understanding. The theory models a gas as a large number of submicroscopic particles in random motion. The derivation involves:
- Calculating the pressure (P) exerted by a single gas molecule colliding with the container walls.
- Averaging this effect for all molecules to relate pressure to the average kinetic energy of the molecules. This yields the kinetic gas equation: P = (1/3)ρv², where ρ is density and v² is the mean square velocity.
- Recognizing that the absolute temperature (T) of a gas is directly proportional to the average kinetic energy of its molecules.
- By substituting and rearranging these relationships, the macroscopic equation PV = nRT is derived, connecting the microscopic behavior (kinetic energy) to the macroscopic properties (P, V, T).
7. Why is the universal gas constant 'R' considered 'universal'?
The gas constant 'R' is called the universal gas constant because it holds the same value for one mole of any gas that behaves ideally, regardless of its chemical identity. Whether it's hydrogen, oxygen, or helium, as long as it follows the ideal gas assumptions, the relationship between its pressure, volume, temperature, and moles will be governed by this single constant. This universality makes 'R' a fundamental constant in physics and chemistry, unlike specific gas constants which are unique to each gas.
