How Is Pressure of an Ideal Gas Calculated?
Before we learn how to calculate the pressure of an ideal gas let us first know what exactly an ideal gas is. An ideal gas in simple words is a theoretical gas in which the gas particles move randomly and there is no interparticle interaction. An ideal gas doesn't exist in reality. It follows the ideal gas equation which is a simplified equation we will learn further and is susceptible to analysis under statistical mechanics. At standard pressure and temperature conditions, most gases are taken to behave as an ideal gas. As defined by IUPAC, 1 mole of an ideal gas has a capacity of 22.71 litres at standard temperature and pressure.
Failure of Ideal Gas Model
At lower temperatures and high pressure, when intermolecular forces and molecular size become important the ideal gas model tends to fail. For most of heavy gases such as refrigerants and gases with strong intermolecular forces, this model tends to fail. At high pressures, the volume of a real gas is often considerably larger than that of an ideal gas and at low temperatures, the pressure of a real gas is often considerably less than that of an ideal gas. At some point in low temperature and high-pressure real gases undergo phase transition which is not allowed in the ideal gas model. The deviation from the ideal gas model can be explained by a dimensionless quantity, called the compressibility factor (Z).
Ideal Gas Equation
Ideal gas law gives an equation known as the ideal gas equation which is followed by an ideal gas. It is a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas equation in empirical form is given as
PV=nRT
where P= pressure of the gas (pascal)
V= volume of gas (liters)
n= number of moles of gas (moles)
R= universal or ideal gas constant (\[=8.314JK^{-1}mol^{-1}\])
T= absolute temperature of the gas (Kelvin)
Ideal gas law is an extension of experimentally discovered gas laws. It is derived from Boyle's law, Charles law, Avogadro's law. When these three are combined, we get an ideal gas law.
Boyle's law => PV = k
Charle's law => V = kT
Avogadro's law => V = kn
Now, when we combine these three laws we use the proportionality constant 'R', which is the universal gas constant and we get the ideal gas equation as
V = RTn/P
=> PV = nRT
Ideal Gas Model Assumptions
Various assumptions are made in the ideal gas model. They are as follows:
Gas molecules are considered as indistinguishably very small and hard spheres.
All motions are frictionless and the collisions are elastic, that is there is no energy loss in motion or collisions.
All laws of Newton are applicable.
The size of the molecules is much smaller than the average distance between them.
There is a constant movement of molecules in random directions with distributed speeds.
Molecules don't attract or repel each other apart from point-like collisions with the walls.
No long-range forces exist between molecules of the gas and surroundings.
The Pressure of an Ideal Gas: Calculation
For the calculation let us consider an ideal gas filled in a container cubical in shape. One corner of the container is taken as the origin and the edges as x, y, and z axes. Let \[A_{1} and A_{2}\] be the parallel faces of the cuboid which are perpendicular to the x-axis. Suppose, a molecule is moving with velocity 'v' in the container and the components of velocity along three axes are \[V_{x}, V_{y} and V_{z}\]. As we assume collisions to be elastic when this molecule collides with face \[A_{1}\] x component of velocity reverses while the y and z component remains unchanged.
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Change in the momentum of the molecule is
\[\Delta P= -mv_{x} -mv_{x} = -2mv_{x} ..... (1)\]
The change in momentum of the wall is \[2mv_{x}\] as the momentum remains conserved.
After the collision, the molecule travels towards the face \[A_{2}\] with the x component of the velocity equal to \[-v_{x}\]
Now, the distance traveled by a molecule from \[A_{1}\] to \[A_{2}\] = L
Therefore, time = \[ \frac{L}{v_{x}}\]
After a collision with \[A_{2}\] it again travels to \[A_{1}\]. Hence, the time between two collisions= \[ \frac{2L}{v_{x}}\]
So the number of collisions of molecule per unit time \[n = \frac{v_{x}}{2L}\]……….(2)
From (1) and (2),
Momentum imparted to the molecule by the wall per unit time
∆F=n∆P
\[ = \sum \frac{m}{L * v_{x}^{2}}\]
Therefore, the total force on wall \[A_{1}\] due to all the molecules is
\[F = \sum \frac{m}{L * v_{x}^{2}}\]
\[F = \frac{m}{L * \sum v_{x}^{2}}\]
\[\sum v_{x}^{2} = \sum v_{y}^{2} = \sum v_{z}^{2} (symmetry)\]
= \[\frac{1}{3}\sum V^{2}\]
Therefore, \[ F = \frac{m}{L*\frac{1}{3N}\sum V^{2}}\]
Now, the pressure is the force per unit area hence,
\[P = \frac{F}{L^{2}}\]
\[\frac{m}{L^{3}(\frac{1}{3N})\sum V^{2}}\]
\[P = \frac{3\rho }{v^{2}}\]
Here, M=total mass of the gas
And ρ=density of the gas
Now, \[\frac{\sum v^{2}}{N}\] is written as \[{v^{2}}\] and is called mean square speed.
\[P = \frac{3\rho }{v^{2}}\]
So, this is what the pressure exerted by gas.
FAQs on Pressure of an Ideal Gas: Laws, Formulas & Uses
1. What is the definition of an ideal gas and how does it differ from a real gas?
An ideal gas is a theoretical gas composed of particles that move randomly and have no intermolecular interactions. It perfectly obeys the gas laws under all conditions of temperature and pressure. A real gas, on the other hand, deviates from ideal behaviour because its particles have finite volume and exert intermolecular forces on each other. The key differences are:
- Particle Volume: Ideal gas particles are considered point masses with zero volume, while real gas particles have a definite, non-zero volume.
- Intermolecular Forces: There are no attractive or repulsive forces between ideal gas particles. Real gases experience weak intermolecular forces (like van der Waals forces).
- Governing Equation: Ideal gases follow the ideal gas equation (PV = nRT) precisely, whereas real gases are better described by equations like the van der Waals equation.
- Conditions for Ideality: Real gases behave most like ideal gases at low pressure and high temperature.
2. What is the formula for the pressure of an ideal gas based on the kinetic theory of gases?
Based on the kinetic theory, the pressure (P) exerted by an ideal gas is derived from the collisions of its molecules with the container walls. The formula is given by:
P = (1/3)ρv²rms
Where:
- P is the pressure of the gas.
- ρ (rho) is the density of the gas (mass per unit volume).
- v²rms is the mean square speed of the gas molecules.
This equation establishes a direct link between the macroscopic property of pressure and the microscopic motion of gas molecules.
3. How is the pressure of an ideal gas related to its volume, temperature, and amount through the Ideal Gas Law?
The relationship between the pressure (P), volume (V), absolute temperature (T), and the number of moles (n) of an ideal gas is described by the Ideal Gas Law. The equation is:
PV = nRT
Here, R is the universal gas constant. This equation shows that:
- Pressure is inversely proportional to volume (at constant temperature and amount).
- Pressure is directly proportional to the absolute temperature (at constant volume and amount).
- Pressure is directly proportional to the amount of gas (n) (at constant volume and temperature).
4. What are some real-world examples where the principles of ideal gas pressure are applied?
The principles governing the pressure of an ideal gas have many practical applications, as most gases at atmospheric conditions behave nearly ideally. Examples include:
- Inflating Tyres: When air is pumped into a tyre, the number of gas molecules increases, leading to more frequent collisions with the inner wall, thus increasing the pressure.
- Aerosol Cans: An aerosol can contains a propellant gas under high pressure. Pressing the nozzle releases the gas, which expands rapidly, forcing the product out.
- Airbags in Vehicles: In a collision, a chemical reaction rapidly produces a large volume of gas (like nitrogen), which inflates the airbag almost instantly due to the immense pressure created.
- Refrigerators and Air Conditioners: These appliances work by compressing and expanding a coolant gas, manipulating its pressure and temperature to transfer heat.
5. What are the key assumptions of the kinetic theory that allow for the derivation of the pressure of an ideal gas?
The derivation of the pressure formula for an ideal gas relies on several key assumptions about the behaviour of gas molecules. These assumptions simplify the model and are crucial for the calculation:
- The gas consists of a large number of identical molecules that are in constant, random motion.
- The volume of the molecules themselves is negligible compared to the volume of the container.
- There are no intermolecular forces of attraction or repulsion between the molecules.
- Collisions between molecules and with the walls of the container are perfectly elastic, meaning kinetic energy is conserved.
- The duration of a collision is negligible compared to the time between collisions.
6. How does the root-mean-square (RMS) speed of gas molecules influence the pressure the gas exerts?
The root-mean-square (RMS) speed is a measure of the average speed of gas particles. It has a direct and significant impact on pressure. According to the kinetic theory formula, P = (1/3)ρv²rms, the pressure is directly proportional to the mean square speed of the molecules. This means that if the RMS speed of the gas molecules increases (for instance, due to a rise in temperature), the molecules will collide with the container walls more forcefully and more frequently, resulting in a higher pressure, assuming the density remains constant.
7. How can we derive the expression for the pressure exerted by an ideal gas in a container?
The expression for pressure is derived by considering the momentum change of gas molecules colliding with a wall of a container. The key steps are:
1. Consider one molecule: A single molecule with velocity component vₓ moves towards a wall of area A. The change in its momentum upon elastic collision is 2mvₓ.
2. Calculate collision frequency: The time between two consecutive collisions with the same wall (in a cubical container of side L) is 2L/vₓ. The frequency of collisions is vₓ/2L.
3. Calculate force by one molecule: The force exerted by one molecule on the wall is the rate of change of momentum, which is (2mvₓ) / (2L/vₓ) = mvₓ²/L.
4. Sum for all molecules: The total force is the sum of forces from all N molecules: F = (m/L) * Σvₓ².
5. Use symmetry: Due to random motion, Σvₓ² = Σvᵧ² = Σv₂² = (1/3)Σv². The total force becomes F = (1/3) * (Nm/L) * v²rms.
6. Calculate Pressure: Pressure P = Force/Area = F/L². This simplifies to P = (1/3)(Nm/L³)v²rms, which is P = (1/3)ρv²rms, since density ρ = Nm/V and V = L³.
8. When solving problems using the Ideal Gas Law, how do I choose between the values R = 8.314 J/(mol·K) and R = 0.0821 L·atm/(mol·K)?
The choice of the value for the universal gas constant (R) depends entirely on the units used for pressure and volume in the problem. There is no other difference between them. The rule is simple:
- Use R = 8.314 J/(mol·K) when pressure is in Pascals (Pa) and volume is in cubic meters (m³). This is the standard SI unit value.
- Use R = 0.0821 L·atm/(mol·K) when pressure is in atmospheres (atm) and volume is in litres (L).
Always check the units of the given quantities in a problem before selecting the value of R to ensure your calculation is correct. Using the wrong value is a very common error.
