# Derivation of Ideal Gas Equation

The ideal gas law is a well-defined approximation of the behaviour of several gases under various situations in thermodynamics. The Ideal Gas Equation is a mathematical formula that uses a combination of empirical and physical constants to express the states of hypothetical gases. The general gas equation is another name for it. Charle’s law, Boyle’s law, Gay-Lussac’s law, and Avogadro’s law are all empirical laws that make up the Ideal Gas Equation. The ideal gas laws and equations are described further below.

### What is an Ideal Gas?

In reality, ideal gas does not exist. It’s a hypothetical gas that’s been proposed to make the calculations easier.

A theoretical gas made up of a collection of randomly moving point particles that only interact through elastic collisions is known as an

ideal gas. The gas molecules in an ideal gas travel freely in all directions, and collisions between them are considered fully elastic, implying that no kinetic energy is lost as a result of the collision.

Although there is no such thing as an ideal gas, when the density is low enough, all real gases tend to approach that characteristic. This is achievable because the gas molecules are spaced so widely apart that they do not interact with one another. Since it obeys the ideal gas law, a simplified equation of state, and is susceptible to statistical mechanics analysis, the ideal gas concept is valuable. As a result, the ideal gas notion aids us in our research.

### Ideal Gas Laws

Certain generalizations have been drawn from the study of gas behaviour. The term “gas laws” refers to these broad generalizations. Ideal gas laws are, as the name implies, laws that deal with ideal gases. The equation of the state of a hypothetical ideal gas is known as the ideal gas law. Although it has significant drawbacks, it provides a decent approximation of the behaviour of various gases under many conditions. When the other two variables are held constant, these laws provide quantitative relationships between any two variables. Let’s have a look at the various gas laws.

**Boyle’s Law (Pressure-Volume Relationship)**

Robert Boyle investigated changes in the volume of a gas by altering the pressure of a specific amount of gas at a constant temperature. He caught some air in the tube’s tip and calculated the pressure exerted by the gas by measuring the difference in mercury height between the tube’s two arms. By adding more mercury to the tube, the pressure of the gas is increased, and the volume of the gas is reduced.

Boyle investigated the relationship between pressure and volume of a given mass of gas at a constant temperature in this way. Boyle’s law is the name given to this relationship.

It asserts that the pressure of a fixed amount of gas varies inversely with the volume of gas at a constant temperature.

It can be expressed as,

**P α 1/V**

where n and T are constant

**P = k _{1}/V **

where k_{1} is a constant of proportionality

**PV=k _{1}**

As a result, Boyle’s law can be written as follows, ”for a given mass of gas at a constant temperature, the product of volume and pressure is constant”.

**Charles’s Law (Volume-Temperature Relationship)**

Jacques Charles investigated the impact of temperature on gas volume at constant pressure in 1787. Gay Lussac expanded on the research in 1802. Charle’s law is a generalization concerning the connection between pressure and volume of a gas that has been observed.

It can be expressed as follows:

“The volume of a fixed mass of gas reduces when it is cooled and increases when the temperature is raised. The volume of the gas grows by 1/273 of its original volume at 0

^{0}C for every degree increase in temperature. Let V_{0}and V_{t}be the volume of the gas at 0^{0}C and t^{0}C, respectively.”

Then,

V_{t}= V_{0}+V_{0 }(t/273.15) ……….. (1)

V_{t}= V_{0}(1+t/273.15) ……….. (2)

V_{t}= V_{0}((273.15+t)/273.15) ………… (3)

We’ll now establish a new temperature scale, with t = T -273.15 for temperature in Celsius and To = 273.15 for temperature in Fahrenheit. The Kelvin temperature scale, often known as the Absolute temperature scale, is a new temperature scale (T). When writing a temperature on the Kelvin scale, the degree sign is omitted. As a result, while writing temperature on the Kelvin scale, we add the temperature in Celsius by 273 to get the Kelvin scale.

Let us assume T_{t} = 273.15 + t

T_{0} = 273.15

The equation (3) can be written as

V_{t}= V_{0}(T_{t}/T_{0})

Or, (V_{t}/V_{0})= (T_{t}/T_{0})

In general, it can be written as,

V_{2}/V_{1}= T_{2}/T_{1}

Or, (V_{1}/T_{1})= (V_{2}/T_{2})

⇒V/T= constant= k_{2}

Hence,

**V=k _{2}T**

where k_{2} is the constant of proportionality.

The volume of a fixed mass of a gas is directly proportional to the absolute temperature when the pressure remains constant, according to Charle’s law.

V α T

**Gay Lussac’s Law**

Joseph Gay Lussac established the relationship between pressure and temperature, which is known as Gay Lussac’s law. It claims that the pressure of a fixed amount of gas varies directly with the temperature at constant volume.

It can be expressed as,

**P α T**

**P = k _{3}T**

(where k_{3} is a constant of proportionality)

**P/T = k _{3}**

As a result, at constant volume, the pressure drops as the temperature decreases, and the pressure rises as the temperature increases.

**Avogadro’s Law **

In 1811, Amadeo Avogadro proposed a formula for calculating the volume of a gas based on the number of molecules present at constant temperature and pressure. The Avogadro law is the name for this. This law asserts that the volume of a gas is directly proportional to the amount of gas at a constant temperature and pressure.

It can be expressed as,

V α n

or

V = k_{4}n

where k_{4} is a constant of proportionality

The amount of gas is represented by the number n. The Avogadro constant is the number of gas molecules in one mole of gas, which has been calculated to be 6.022×10^{23}.

### The Combined Gas Law or Ideal Gas Equation

Boyle’s law and Charle’s law both give variations in the volume of a gas as a function of pressure and temperature. We can construct an equation that shows the simultaneous effect of pressure and temperature changes on the volume of a gas by combining these two rules. The combined gas law, often known as the ideal gas equation.

**Ideal Gas Equation**

The equation for the state of a hypothetical ideal gas is known as the ideal gas law. Although it has significant drawbacks, it is a good approximation of the behaviour of various gases under many conditions.

The ideal gas equation is stated as

PV = nRTwhere,

P is the ideal gas’s pressure.

The volume of the ideal gas is V.

The amount of ideal gas, measured in moles, is n.

The universal gas constant is R.

T stands for temperature.

The product of a gas’s pressure and volume has a constant relationship with the product of a universal gas constant and temperature, according to the Ideal Gas Equation.

## Derivation of the Ideal Gas Equation

Assume that the pressure exerted by the gas is ‘P.’ ‘V’ is the volume of the gas. ‘T’ is the temperature. ‘n’ is the number of moles of gas.

According to Boyle’s Law, the volume of a gas is inversely proportional to the pressure exerted by it at constant n and T.

V ∝ 1/P ………………(1)

When P and n are constant, the volume is directly proportional to the temperature, according to Charles’ Law.

V ∝ T ……………..(2)

Avogadro’s Law asserts that the volume of a gas is directly proportional to the number of moles of gas when P and T are constant.

V ∝ n …………………(3)

When all three equations are combined, we get,

V ∝ nT/P

PV ∝ nT

PV = nRTwhere R is the Universal gas constant, which is 8.314 J/mol-K.

### Sample Questions

**Question 1: What does the ideal gas equation imply?**

**Answer:**

The product of the pressure and volume of one mole of a gas is equal to the product of its temperature and the gas constant in this equation. For an ideal gas, the equation is precise, and for real gases at low pressures, it is a decent approximation.

**Question 2: What is a good example of an ideal gas?**

**Answer:**

Many gases, including nitrogen, oxygen, hydrogen, noble gases, some heavier gases like carbon dioxide, and mixtures like air, can be regarded as ideal gases within reasonable tolerances throughout a wide temperature and pressure range.

**Question 3: What is the temperature, at 1.00 atm pressure in kelvin, one mole of CH _{4} gas that occupies 20.0L?**

**Answer:**

The ideal gas equation is,

PV=nRT

T=PV/nR ………(1)

Given 1.00 atm pressure, so P=1.00 atm. one mole of CH

_{4}gas, so n=1mol, the CH gas occupies 20.0L, so V=20.0L. The gas constant is R=0.082.On substituting these value sin equations (1).

T = (1.00atm)(20.0L)/(1mol)(0.082)

T = 244K

**Question 4: At the same temperature and pressure, which is denser, dry air or air saturated with water vapour?**

**Answer:**

Since it has a greater molar mass, air saturated with water vapor is denser at the same temperature and pressure.

**Question 5: Let a gas with a standard temperature and pressure undergoes a modification that reduces its pressure by half. During this process, how much does the volume of the gas change?**

**Answer:**

The ideal gas equation is,

PV=nRT

The given gas undergoes a transformation in which its pressure is half. Therefore:

P′=P/2

The ideal gas equation can also be written as,

V=nRT/P

and

V′=nRT/P′ ……….(1)

On substituting the value of P’ in equation (1).

V’=(nRT)/(P/2)

V’=2(nRT/P)

V’=2V

As a result, we can observe that the new volume is twice the original volume.

**Question 6: What do you consider to be ideal gas conditions?**

**Answer:**

The following are the basic assumptions for a gas to be ideal:

- The volume of the gas particles is insignificant.
- The gas particles are all the same size and there are no intermolecular forces (attraction or repulsion) between them.
- Perfect elastic collisions occur between the gas particles, with no energy loss.

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