Ideal Gas Law | Brilliant Math & Science Wiki (2024)

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Rohit Gupta, Akshay Yadav, July Thomas, and

  • Sravanth C.
  • Abhiram Rao
  • Anik Mandal
  • Eli Ross

contributed

An ideal gas is a theoretical gas composed of many randomly moving point particles that do not interact except when they collide elastically. The ideal gas law is the equation of state of an ideal gas. It relates the state variables of the gas: pressure \((P),\) volume \((V),\) and temperature \((T).\) Also included are the amount of the gas \((n)\) and the ideal gas constant \((R=8.314 \frac{\text{J}}{\text{K mol}}).\)

\[PV=nRT\]

Contents

  • Kinetic Theory of ideal gases (Assumption for Ideal gases)
  • Gay-Lussac's law or Amontons' Law
  • Boyle's Law
  • Charles' Law
  • Avogadro's Hypothesis
  • Dalton's Law of Partial Pressures
  • Ideal gas Equation

Kinetic Theory of ideal gases (Assumption for Ideal gases)

To describe an ideal gas, a set of assumptions are made.

1) Gases consist of large numbers of tiny particles that are far apart relative to their size. This implies that the gas molecules have negligible volume compared to the volume of container in which they are placed.

2) Collisions between gas particles and between particles and container walls are elastic collisions (there is no net loss of total kinetic energy).

3) Gas particles are in continuous, rapid and random motion. They therefore possess kinetic energy, which is energy of motion.

4) There are no forces of interaction between gas particles. Thus they can move independent of each other. They only interact with each other through elastic collisions.

5) The average kinetic energy of a gas particle depends only on the temperature of the gas.

Gay-Lussac's law or Amontons' Law

The last postulate of the kinetic molecular theory states that the average kinetic energy of a gas particle depends only on the temperature of the gas. Thus, the average kinetic energy of the gas particles increases as the gas becomes warmer. Because the mass of these particles is constant, their kinetic energy can only increase if the average velocity of the particles increases. The faster these particles are moving when they hit the wall, the greater the force they exert on the wall. Since the force per collision becomes larger as the temperature increases, the pressure of the gas must increase as well.\[P \propto T\]
Here P is pressure and T is temperature in kelvin.In this, volume and number of moles of gas is taken constant.If temperature is represented in kelvin then the graph between pressure and temperature will be a straight line passing through origin.
Ideal Gas Law | Brilliant Math & Science Wiki (1)

If temperature is represented in celsius then the graph between pressure and temperature will be a straight line but will not pass through origin. On extrapolating, the graph will hit -273.15 degrees.

Ideal Gas Law | Brilliant Math & Science Wiki (2)

Boyle's Law

Gases can be compressed because most of the volume of a gas is empty space. If we compress a gas without changing its temperature, the average kinetic energy of the gas particles stays the same. There is no change in the speed with which the particles move, but the container is smaller. Thus, the particles travel from one end of the container to the other in a shorter period of time. This means that they hit the walls more often. Any increase in the frequency of collisions with the walls must lead to an increase in the pressure of the gas. Thus, the pressure of a gas becomes larger as the volume of the gas becomes smaller.

If temperature and amount of gas is fixed then pressure is inversely proportional to volume occupied by the gas.
\[P \propto 1/V\]
Here P is pressure and V is volume.

If temperature and number of moles of gas are fixed then the graph between pressure P and volume V will be a rectangular hyperbola. On increasing volume of gas pressure decrease and vice-versa. Such a process is also called as Isothermal Process
Ideal Gas Law | Brilliant Math & Science Wiki (3)

Charles' Law

The average kinetic energy of the particles in a gas is proportional to the temperature of the gas. Because the mass of these particles is constant, the particles must move faster as the gas becomes warmer. If they move faster, the particles will exert a greater force on the container each time they hit the walls, which leads to an increase in the pressure of the gas. If the walls of the container are flexible, it will expand until the pressure of the gas once more balances the pressure of the atmosphere. The volume of the gas therefore becomes larger as the temperature of the gas increases.

If pressure of an ideal gas is kept constant then volume of container is directly proportional to temperature (in kelvin) of the gas.
\[V \propto T\]
Here, V is volume of container and T is temperature of gas in Kelvin.

The graph between V and T (in kelvin) depicting the Charles' law will be a straight line passing through the origin. Although, we can never reduce the volume to zero thus the graph should not be shown passing through the origin.
Ideal Gas Law | Brilliant Math & Science Wiki (4)

If the temperature is taken in degrees then the graph will be instead of passing through origin will hit temperature \(-273^\circ \)C Ideal Gas Law | Brilliant Math & Science Wiki (5)

Avogadro's Hypothesis

As the number of gas particles increases, the frequency of collisions with the walls of the container must increase. This, in turn, leads to an increase in the pressure of the gas. Flexible containers, such as a balloon, will expand until the pressure of the gas inside the balloon once again balances the pressure of the gas outside. Thus, the volume of the gas is proportional to the number of gas particles.

If pressure and temperature of an ideal gas is kept constant then volume of container is directly proportional to the amount of gas (number of moles of gas) in the container.
\[V \propto N\]

Ideal Gas Law | Brilliant Math & Science Wiki (6)

Dalton's Law of Partial Pressures

Imagine what would happen, gases at different pressure but same temperature are added to a container. The total pressure would increase because there would be more collisions with the walls of the container. There is so much empty space in the container that each type of gas molecules hits the walls of the container as often in the mixture as it did when there was only one kind of gas. The total pressure will increase as more number of gas molecules hits the container walls but the pressure due to individual gas molecules remains same. The total number of collisions with the wall in this mixture is therefore equal to the sum of the collisions that would occur when each gas is present by itself. In other words,

The total pressure of a mixture of gases is equal to the sum of the partial pressures of the individual gases.
\[{P_t} = {P_1} + {P_2} + {P_3} + ...\]

Ideal gas Equation

It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law, Charles's law and Avogadro's Law.The ideal gas law is often written as:

\(PV=nRT \)

Where, \(R\) is the Gas Constant. Some values of \(R\) are given below.

\(R=8.314\) \(\text{J.mol}^{-1}\text K^{-1}\)

\(R=0.082\) \(\text{litre.atm.mol}^{-1}\text K^{-1}\)

\(R=8.2057\) \(\text m^{3}\text{.atm.mol}^{-1}\text K^{-1}\)

A gas at 27°C has a volume V and pressure P. On heating its pressure is doubled and volume becomes three times. Find the resulting temperature of the gas.

From ideal gas equation,\[PV = nRT \]For a closed container, number of moles remains constant, therefore
\[\frac{{{T_2}}}{{{T_1}}} = \left( {\frac{{{P_2}}}{{{P_1}}}} \right)\,\left( {\frac{{{V_2}}}{{{V_1}}}} \right) = \left( {\frac{{2{P_1}}}{{{P_1}}}} \right)\,\left( {\frac{{3{V_1}}}{{{V_1}}}} \right) = 6\]
\[{T_2} = 6{T_1} = 6 \times 300\]
\[{T_2} = 1800\,K = 1527^\circ C.\]

A balloon contains \(500\,{m^3}\) of helium at 27°C and 1 atmosphere pressure. The volume of the helium at – 3°C temperature and 0.5 atmosphere pressure will be

From Ideal gas equation,\[PV = nRT \]

Since gas is trapped inside balloon therefore number of moles of gas remains unchanged.Let \({P_1}\), \({V_1}\) and \({T_1}\) are initial pressure, volume and temperature, and \({P_2}\), \({V_2}\) and \({T_2}\) are final pressure, volume and temperature
then,

\[\frac{{{V_2}}}{{{V_1}}} = \left( {\frac{{{T_2}}}{{{T_1}}}} \right)\,\left( {\frac{{{P_1}}}{{{P_2}}}} \right) = \left( {\frac{{270}}{{300}}} \right)\,\left( {\frac{1}{{0.5}}} \right) = \frac{9}{5}\]

\[{V_2} = 500 \times \frac{9}{5} = 900\,{m^3}\]

An ideal gas initially at \(27^\circ\text{C}\) is kept at a constant volume and pressure as \( \dfrac23\) of the gas is released to the surroundings. What is the new temperature of the gas in \(^\circ\text{C}?\)

Cite as: Ideal Gas Law. Brilliant.org. Retrieved from https://brilliant.org/wiki/ideal-gas-law/

Ideal Gas Law | Brilliant Math & Science Wiki (2024)

FAQs

What is the ideal gas law in math? ›

As the different pieces of this puzzle came together over a period of 200 years, we arrived at the ideal gas law, PV=nRT, where P is pressure, V is volume, T is temperature, n is # of molecules and R is the universal gas constant.

What is the main idea of the ideal gas law? ›

The ideal gas law states that the product of the pressure and the volume of one gram molecule of an ideal gas is equal to the product of the absolute temperature of the gas and the universal gas constant.

What is the best mathematical statement of the ideal gas law? ›

In such a case, all gases obey an equation of state known as the ideal gas law: PV = nRT, where n is the number of moles of the gas and R is the universal (or perfect) gas constant, 8.31446261815324 joules per kelvin per mole.

What best summarizes the ideal gas law? ›

So, in summary, the Ideal Gas Law states that under the same temperature, pressure and volume all gases contain the same number of molecules (but not the same mass).

What is the ideal gas law in real life? ›

Ideal Gas law has a lot more practical applications. It is being used to determine the densities of gases and in stoichiometric calculations. The coolants/refrigerants in your refrigerator, hot air balloons in the sky, and combustion engines in vehicles, all are based on the ideal gas law.

Does ideal gas exist? ›

An ideal gas is purely hypothetical and such gases do not exist in reality. Q.

Why is the ideal gas law useful? ›

Gases are complicated. They're full of billions and billions of energetic gas molecules that can collide and possibly interact with each other. Since it's hard to exactly describe a real gas, people created the concept of an Ideal gas as an approximation that helps us model and predict the behavior of real gases.

What is ideal gas theory? ›

An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.

What does R mean in ideal gas law? ›

The term "R" in the equation PV=n RT stands for the universal gas constant. R = PV/ nT. The universal gas constant is a proportionality constant that relates the kinetic energy of a sample of gas to its temperature and molarity.

What is an interesting fact about the ideal gas law? ›

Ideal gas molecules themselves take up no volume.

In fact, for temperatures near room temperature and pressures near atmospheric pressure, many of the gases we care about are very nearly ideal.

Who invented the ideal gas law? ›

The ideal gas law was discovered by physicist and engineer Benoît Paul Émile Clapeyron (seen on the right) in 1834.

Is an ideal gas a hypothetical gas? ›

An ideal gas is defined as a hypothetical gaseous substance whose behavior is independent of attractive and repulsive forces and can be completely described by the ideal gas law.

What best describes the ideal gas law? ›

The ideal gas law is expressed as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the absolute temperature. At 0°C and 1 atm, gases that behave most ideally are those that have relatively low molecular masses and weak intermolecular forces.

What does n mean in ideal gas law? ›

The ideal gas law can also be written and solved in terms of the number of moles of gas: PV = nRT, where n is number of moles and R is the universal gas constant, R = 8.31 J/mol ⋅ K.

What gas acts the most ideal? ›

The real gas that acts most like an ideal gas is helium. This is because helium, unlike most gases, exists as a single atom, which makes the van der Waals dispersion forces as low as possible. Another factor is that helium, like other noble gases, has a completely filled outer electron shell.

What is the ideal gas equation in simple terms? ›

Ideal Gas Equation is the combination of empirical laws like Charle's law, Boyle's law, Gay-Lussac's law, and Avogadro's law. Ideal Gas Equation is the equation defining the states of the hypothetical gases expressed mathematically by the combinations of empirical and physical constants.

What is the ideal gas equation answer? ›

The ideal gas equation is formulated as: PV = nRT. In this equation, P refers to the pressure of the ideal gas, V is the volume of the ideal gas, n is the total amount of ideal gas that is measured in terms of moles, R is the universal gas constant, and T is the temperature.

What is the ideal real gas equation? ›

Originally, the ideal gas law looks like this: PV = nRT. P is the pressure in atmospheres, V is the volume of the container in liters, n is the number of moles of gas, R is the ideal gas constant (0.0821 L-atm/mol-K), and T is the temperature in Kelvin.

What is the ideal gas law expressed as PV nRT? ›

Definition. The Ideal Gas law (PV = nRT) is an equation representing the state of a hom*ogenous mixture of gas, which sets variables of that gas's pressure (P) times volume (V) equal to the amount in moles (n) of that gas multiplied by the ideal gas constant (R) multiplied by its temperature (T).

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