## Ideal Gas Model

The** ideal gas model** is used to predict the behavior of gases and is one of the most useful and commonly used substance models ever developed. I was found, that if we confine **1 mol samples** of **various gases** in **identical volume** and hold the gases at the **same temperature**, then their measured **pressures are almost the same**. Moreover when we confine gases at lower densities the differences tend to disappear. It was found, such gases tend to obey the following relation, which is known as the **ideal gas law**:

*pV = nRT*

where:

** p** is the

**absolute pressure**of the gas

** n** is the

**amount**of substance

** T** is the

**absolute temperature**

** V **is the

**volume**

** R **is the ideal, or universal,

**gas constant**, equal to the product of the

**Boltzmann constant**and the

**Avogadro constant.**The power of the

**ideal gas law**is in its simplicity. When any two of the thermodynamic variables, p, v, and T are given, the third can easily be found.

The** ideal gas model** is based on following assumptions:

- The pressure, volume, and temperature of an ideal gas obey the
**ideal gas law**. - The
**specific internal energy**is only a function of the temperature:*u = u(T)* - The molar mass of an ideal gas is identical with the molar mass of the real substance
- The
**specific heats**—and*c*_{p}— are independent of temperature which means that they are constants.*c*_{v}

From microscopic point of view it is based on these assumptions:

- The molecules of the gas are
**small, hard spheres**. - The only forces between the gas molecules are those that determine the
**point-like collisions**. - All collisions are
**elastic**and all motion is**frictionless**. - The average distance between molecules is much larger than the size of the molecules.
- The molecules are moving in random directions.
- There are no other attractive or repulsive force between these molecules.

## What is an Ideal Gas

An **ideal gas** is defined as one in which all collisions between atoms or molecules are **perfectly elastic** and in which there are **no intermolecular attractive forces**. An ideal gas can be visualized as a collection of perfectly hard spheres which collide but which otherwise do not interact with each other. In reality, no real gases are like an ideal gas and therefore no real gases follow the** ideal gas law** or equation completely. At temperatures near a gases boiling point, increases in pressure will cause condensation to take place and drastic decreases in volume. At very high pressures, the intermolecular forces of a gas are significant. However, most gases are in approximate agreement at pressures and temperatures above their boiling point. The **ideal gas law** is utilized by engineers working with gases because it is **simple to use** and approximates real gas behavior.

See also: Elastic Collision

## Joule’s Second Law

For any gas whose equation of state is given exactly by * pV = nRT *(or

*), the*

**pv = RT****specific internal energy**depends on temperature only. This rule was originally found in 1843 by an English physicist

**James Prescott Joule**experimentally for real gases and is known as

**Joule’s second law**:

*The internal energy of a fixed mass of an ideal gas depends only on its temperature (not pressure or volume).*

The specific enthalpy of a gas described by * pV = nRT* also depends on temperature only. Note that, the

**enthalpy**is the thermodynamic quantity equivalent to the

**total heat content**of a system. It is equal to the internal energy of the system plus the product of pressure and volume. In intensive variables the

**Joule’s second law**is therefore given by

*h = h(T) = u(T) + pv = u(T) + RT.*

These three equations constitute the ideal gas model, summarized as follows:

**pv = RT**

**u = u(T)**

**h = h(T) = u(T) + RT**

## Ideal Gas Law

Any equation that relates the pressure, temperature, and specific volume of a substance is called an **equation of state**. The simplest and **best-known** equation of state for substances in the gas phase is the **Ideal Gas equation** of state. It was first stated by Émile Clapeyron in 1834 as a combination of the empirical Boyle’s law, Charles’ law and Avogadro’s Law. This equation predicts the **p-v-T behavior** of a gas quite accurately for dilute or low-pressure gases. In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with** temperature** and **quantity**, and inversely with **volume**.

*pV = nRT*

where:

is the*p***absolute pressure**of the gasis the*n***amount**of substanceis the*T***absolute temperature**is the*V***volume**is the ideal, or universal,*R***gas constant**, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the **universal gas constant** that has the same value for all gases—namely, **R = 8.31 J/mol K.**

The power of the ideal gas law is in its **simplicity**. When any** two** of the thermodynamic variables, p, v, and T** are** **given**, the **third** can **easily be found**. Many physical conditions of gases calculated by engineers fit the above description. Perhaps the most common use of gas behavior studied by engineers is that of the compression process and expansion process using ideal gas approximations.

## Gas Laws

In general, the **gas laws** are** first equations of state**, that correlate densities of gases and liquids to temperatures and pressures. The** gas laws** were completely developed at the end of the 18th century. These laws or statements **preceded** the **ideal gas law**, since individually these laws are considered as special cases of the ideal gas equation, with one or more of the variables held constant. Since they have been almost completely replaced by the ideal gas equation, it is not usual for students to learn these laws in detail. The **ideal gas equation** was first stated by Émile Clapeyron in 1834 as a combination of these laws:

## Example: Ideal Gas Law – Gas compression inside a pressurizer

**Pressure in the primary circuit** of PWRs is maintained by a **pressurizer**, a separate vessel that is connected to the primary circuit (hot leg) and partially filled with water which is heated to the **saturation temperature** (boiling point) for the desired pressure by submerged **electrical heaters**. During the plant heatup the pressurizer can be filled by nitrogen instead of saturated steam.

Assume that a pressurizer contains **12 m**** ^{3}** of nitrogen at

**20°C**and

**15 bar**. The temperature is raised to

**35°C**, and the volume is reduced to

**8.5 m**

**. What is the final pressure of the gas inside the pressurizer? Assume that the gas is ideal.**

^{3}**Solution:**

Since the gas is ideal,we can use the ideal gas law to relate its parameters, both in the **initial state i **and in the **final state f**. Therefore:

*p _{init}V_{init} = nRT_{init}*

and

*p _{final}V_{final} = nRT_{final}*

Dividing the second equation by the first equation and solving for ** p_{f}** we obtain:

*p _{final} = p_{init}T_{final}V_{init} / T_{init}V_{final}*

Note that we cannot convert units of volume and pressure to basic SI units, because they cancel out each other. On the other hand we have to use Kelvins instead of degrees of Celsius. Therefore T_{init} = 293 K and T_{final} = 308 K.

It follows, the resulting pressure in the final state will be:

* p _{final} *

*= (15 bar) x (308 K) x (12 m*

^{3}) / (293 K) x (8.5 m^{3}*) =*

*22 bar*## Validity of Ideal Gas Law

Since **ideal gas** is defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces, there is no such thing in nature as a truly ideal gas. On the other hand, all real gases approach the ideal state **at low pressures (densities)**. At low pressures molecules are far enough apart that they do not interact with one another.

In other words, the **Ideal Gas Law** is accurate only **at relatively low pressures** (relative to the critical pressure p_{cr}) and **high temperatures** (relative to the critical temperature T_{cr}). At these parameters, the **compressibility factor, Z = pv / RT**, is

**approximately 1**. The compressibility factor is used to account for deviation from the ideal situation. This correction factor is dependent on pressure and temperature for each gas considered.

## Internal Energy of an Ideal Gas

The **internal energy** is the total of all the energy associated with the motion of the atoms or molecules in the system. **Microscopic forms** of energy include those due to the **rotation**, **vibration**, **translation**, and** interactions** among the molecules of a substance.

## Monatomic Gas – Internal Energy

For a **monatomic ideal gas** (such as helium, neon, or argon), the only contribution to the energy comes from **translational kinetic energy**. The average translational kinetic energy of a single atom depends **only** on the **gas temperature** and is given by equation:

* K _{avg} = 3/2 kT.*

The internal energy of n moles of an ideal monatomic (one atom per molecule) gas is equal to the average kinetic energy per molecule times the total number of molecules, N:

*E _{int} = 3/2 NkT = 3/2 nRT*

where n is the number of moles. **Each direction** (x, y, and z) contributes ** (1/2)nRT **to the

**internal energy**. This is where the

**equipartition of energy idea**comes in – any other contribution to the energy must also contribute

**. As can be seen, the internal energy of an ideal gas**

*(1/2)nRT***depends only on temperature**and the number of moles of gas.

## Diatomic Molecule – Internal Energy

If the gas molecules contain more than one atom, there are **three translation directions**, and **rotational kinetic energy** also contributes, but only for rotations about two of the three perpendicular axes. The five contributions to the energy (five degrees of freedom) give:

**Diatomic ideal gas:**

*E _{int} = 5/2 NkT = 5/2 nRT*

This is only an approximation and applies at intermediate temperatures.** At low temperatures** only the **translational kinetic energy contributes**, and at higher temperatures two additional contributions (kinetic and potential energy) come from vibration. The **internal energy will be greater** at a given temperature than for a monatomic gas, but it will still be a function only of temperature for an ideal gas.

The internal energy of real gases also depends mainly on temperature, but similarly as the **Ideal Gas Law**, the internal energy of real gases depends also somewhat on **pressure** and **volume**. All real gases approach the ideal state at low pressures (densities). At low pressures molecules are far enough apart that they do not interact with one another. The internal energy of liquids and solids is quite complicated, for it includes **electrical potential energy** associated with the forces (or **chemical bonds**) between atoms and molecules.

## Specific Heat at Constant Volume and Constant Pressure

**Specific heat** is a property related to** internal energy** that is very important in thermodynamics. The **intensive properties c_{v}** and

*are defined for pure, simple compressible substances as partial derivatives of the*

**c**_{p}**internal energy**and

*u(T, v)***enthalpy**, respectively:

*h(T, p)*where the subscripts **v** and **p** denote the variables held fixed during differentiation. The properties **c _{v} **and

**c**are referred to as

_{p}**specific heats**(or

**heat capacities**) because under certain special conditions they relate the temperature change of a system to the amount of energy added by heat transfer. Their SI units are

**J/kg K**or

**J/mol K**. Two specific heats are defined for gases, one for

**constant volume (c**and one for

_{v})**constant pressure (c**.

_{p})According to the **first law of thermodynamics**, for constant volume process with a monatomic ideal gas the molar specific heat will be:

*C _{v} = 3/2R = 12.5 J/mol K*

because

*U = 3/2nRT*

It can be derived that the **molar specific heat** at constant pressure is:

**C _{p} = C_{v} + R = 5/2R = 20.8 J/mol K**

This ** C_{p}** is greater than the molar specific heat at constant volume

**, because energy must now be supplied**

*C*_{v}**not only**to

**raise the temperature**of the gas but also for the

**gas to do work**because in this case volume changes.

## Mayer’s relation – Mayer’s formula

Julius Robert Mayer, a German chemist and physicist, derived a relation between **specific heat at constant** pressure and the **specific heat at constant volume** for an ideal gas. He studied the fact that the specific heat capacity of a gas at constant pressure (C_{p}) is slightly greater than at constant volume (C_{v}). He reasoned that this *C***_{p}** is greater than the molar specific heat at constant volume

*C***, because energy must now be supplied**

_{v}**not only**to

**raise the temperature**of the gas but also for the

**gas to do work**because in this case volume changes. According to the

**Mayer’s relation**or the

**Mayer’s formula**the difference between these two heat capacities is equal to the universal gas constant, thus the molar specific heat at constant pressure is equal:

**C _{p} = C_{v} + R**

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