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In this article we will discuss about:- 1. Ideal Gas Laws 2. Equation of State or Characteristic Gas Equation 3. Universal Gas Constant 4. Joule’s Experiment of Ideal Gases to Prove U = f (T) 5. Relations between Cp and Cv 6. Ideal Gas Processes.

**Contents:**

- Ideal Gas Laws
- Equation of State or Characteristic Gas Equation
- Universal Gas Constant
- Joule’s Experiment of Ideal Gases to Prove U = f (T)
- Relations between Cp and Cv
- Ideal Gas Processes

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**1. Ideal Gas Laws**:

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**The behaviour of ideal gas is governed by following gas laws: **

**1. Boyle’s Law:**

Boyle experimentally established that, the volume of a given mass of a gas is inversely proportional to the absolute pressure, when the temperature is constant.

where C is a constant.

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Also P_{1}V_{1} = P_{2}V_{2} = C

Consider a gas in a cylinder fitted with a frictionless piston. Let the initial properties be P_{1} and V_{1.} Now when the gas is heated at constant temperature, then the gas expands and the pressure falls to P_{2} and volume increases to V_{2}. This is represented on PV-diagram.

**2. Charle’s Laws:**

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(i) It states that, “when the pressure of a given mass of a gas is kept constant, then volume is directly to the absolute temperature”.

(ii) It also states that, “when the volume of a given mass of a gas is kept constant, the pressure is directly proportional to the absolute temperature”.

**3. Avagadro’s Law:**

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This law states that at the same pressure and temperature, equal volumes of different gases contain equal number of molecules.

** 2. Equation of State or Characteristic Gas Equation****: **

In engineering practice, pressure, volume, temperature all vary simultaneously. So, Boyle’s law or Charle’s law alone is not applicable, since one of three properties is kept constant. In order to establish a relationship between these three properties, Boyle’s law and Charle’s laws are combined together, which gives a general equation called characteristic gas equation.

Frequently it is required to express the equation of state, PV = mRT. (1) on mole basis. The mass of a substance m is equal to the product of number of moles n and the molecular weight M.

** 4. Joule’s Experiment of Ideal Gases to Prove U = f (T)****:**

Joule’s Law states that the internal energy of an ideal gas depends only on the temperature of gas and is independent of changes in pressure and volume i.e., U = *f *(T).

**Joule’s Experiment: **

The apparatus for Joule’s experiment is shown in Fig. 4.4. It consists of two copper vessels A and B connected by a valve V. Initially A contained high pressure air and B was evacuated. Both were placed in an insulated water bath with a thermometer to measure the temperature.

The initial temperature of water was recorded. The valve V was then opened and air is allowed to flow from vessel A to vessel B.

The operation is slow enough for air to attain equilibrium states. Eventually pressure in both the vessels will become equal and flow will be stopped. Joule observed that there was no change in the temperature of water during or after the process.

Since there was no change in temperature, there was no heat transfer between air and water. Therefore δQ = 0. Also no work is done during the process. Therefore δW = 0. By first law of thermodynamics for closed system we know that δQ = dU+ δW.

Therefore dU = 0 or change in internal energy is zero. The internal energy therefore remains constant. Again we observe that pressure and volume both changed during the process and only temperature remained constant. Joule therefore concluded that internal energy is the function of temperature only and independent of pressure and volume changes.

Thus, U = *f *(T)

** 5. Relations between**** C _{p} and C_{v}:**

Specific heat at constant pressure and specific heat at constant volume in the I-law of thermodynamics.

Since C_{p} and C_{v} are the properties of a system, there is a relationship between these. We know that, specific enthalpy h is given by,

During this process, volume remains constant (V = C) and is represented on a PV-diagram by means of vertical line as shown.

When unit mass of a gas is heated in a closed vessel (i.e., V = C), then, since volume remains constant, no external work is done. But as temperature of the gas increases I.E. increases.

**(a) Work Done:**

Work done in a non-flow system = ∫ PdV.

Here, V = C

∴ dV = 0

∴ W_{1-2} = 0

**(b) Heat Supplied:**

From the I-Law for a closed system undergoing a process,

During this process pressure remains constant (P = C) and this process is represented by means of a horizontal line on the PV-diagram as shown.

When a unit mass of a gas is taken in a cylinder fitted with a frictionless piston and is heated. Then the piston moves up maintaining same pressure (i.e., P = C). As the volume of the gas increases, the work is done by the gas on the piston. As the temperature of the gas increases, Internal Energy increases.

**So, heat supplied in a constant pressure heating process is utilized for two purposes: **

(i) For doing some external work.

(ii) For increasing the I.E. of the gas.

Work done in non-flow system.

**(b) Change in Internal Energy:**

(ΔU) = U_{2 }– U_{1}

**(c) Heat Transferred: **

From the I-law,

During this process, temperature remains constant (T = C). The law for the process is PV = C and is represented by means of a curve as shown on the PV-diagram. It is represented by means of a horizontal line on T-S diagram.

**(a) Work Done:**

The work done in a non-flow system.

**(b) Change in IE: **

**(c) Heat Transferred: **

**4. Reversible Adiabatic Process****: **

A process is said to be reversible adiabatic, when the heat added or rejected during the process is zero. In this case the process should be very fast so that there is very little time for the exchange of heat to take place. The system for this process may be considered as piston and cylinder mechanism.

We know that first law for the process is,

Now for Adiabatic Process (PV^{γ} = C) and during this process neither heat enters the system nor heat leaves the system,

**(a) Work Done: **

**Note: **

Derivation is just similar to work done in polytrophic process, put γ in place of n.

**(b) Heat Supplied: **

Here Heat supplied = 0 (Since neither heat enters nor leaves the system during adiabatic the process).

Polytropic Process is given by the law (PV^{n} = C) and is represented on the PV-diagrams as shown,

**(a) Work Done: **

Work done in a non-flow process

**(b) Heat Supplied: **

Throttling is an irreversible process in which a fluid, flowing across a restriction, undergoes a drop in a total pressure, such a process occurs in the flow through a porous plug, a partially closed valve, or a small orifice.

Joule and Thomson performed the basic throttling experiments in the period 1852-62, and their experiments clarified the process and let to use of throttling as a method for determining certain properties of gaseous substances.

A steady stream of gas flows through a porous plug contained in a horizontal tube. This system is open, is thermally insulated (Q = 0) and does not exchange work with its environment (W = 0). At sections 1 and 2, both the temperature and the pressure are measured. If the kinetic energy does not change significantly as the fluid passes through the porous plug, the steady flow energy equation reduces to –

Hence, in an adiabatic throttling process the enthalpy remains constant. In actual experiments with different p_{2},_{ }t_{2} does not remain constant but is different than t_{1}.

Table 4.1 shows the various quantities involved in various processes-

Since it is helpful to be able to sketch curves of various thermodynamic process on P-V and T-S planes about as they should appear Fig. 4.12 will be of assistance in learning to make these sketches. It shows also the effect of varying n. Note that the isentropic curve on the PV plane is steeper than the isothermal curve and that, on TS-plane, the constant volume curve is steeper than the constant pressure curve when both the curves are drawn for the same temperature limits.

Expansions or compressions are imagined to take place from some common point’ 1′.