Equation of state of a perfect gas

Introduction

The perfect gas law defines the relationship between pressure, volume, and temperature with a valid formula for natural gases. The law states that there is a direct proportionality between the three variables – if anyone doubles, then so will the others. Moreover, it studies when two quantities are variable, and the third is constant.

1. The ideal gas law is a short statement of the physical behavior of gases. It contains Boyle’s law and Charles’s law as special cases. It expresses that pressure, volume, and temperature can be written as a function of one variable (temperature) only.
2. The general gas law and equation of a perfect gas are derived from the kinetic theory of gases. They are based on assumptions that large numbers of molecules are in motion.
3. The volume occupied by the gas molecules is negligibly small compared to the volume occupied by the gas itself.
4. Additionally, no forces act on the molecules except during elastic collisions of negligible duration.
5. Though many substances have the same observable properties as gases, none of them behave exactly like a gas.

However, the behavior of actual gases is described quite accurately by the ideal gas law  at sufficiently high temperatures and low pressures,  relatively large distances between molecules and their high speeds overcome any interaction they might exert on one another.

Dimensional Analysis of Equation of State of a Perfect Gas

The dimensional analysis of the ideal gas law is important for all students. With the equation as follows: PV=nRT Here, P(Pressure) is with V(Volume), and so the dimensions of pressure are [M L^-1 T^-2], and that of volume is [L³]. So by combining both the measurements, we get PV = [M L² T^-2]. Now, on the right-hand side, n = [mol], R = [M L² T^-2 mol^-1 K^-1] and T = [K]. So, the dimensions are [M L² T^-2], and hence both the sides are equal now. The EquationEquation of the state of perfect gas has the dimensions stated above.

Boyle’s Law

“‘Boyle’s Law” is a thermodynamic law that, with the assumption of an ideal gas, establishes that the pressure and volume of a given amount of gas are inversely proportional. In it, the third quantity, temperature, will be kept constant. In terms of the equation, it can be written as : PV = k, which is a constant Moreover, if the temperature is constant, then the equation goes out as : P₁V₁ = P₂V₂ The graph shows an inverse relationship between pressure and volume at two different temperatures (T₁<T₂)

Charles’s Law

Charles’ law is explained in the following way. As temperature increases, the volume of a given gas mass expands and vice versa, i.e., when temperature decreases, the volume contracts. This law is applicable given that pressures remain constant during any process. In terms of the equation, it can be written as : V/T = constant Moreover, if the pressure is constant, then the equation goes out as : V₁T₂ = V₂T₁ There is a clear deviation between the theoretical and experimental results. The hypothetical graph certainly deviates from the straight line at high temperatures.

Ideal Gas Equation

When we break down the Ideal Gas Law, it’s a simple equation that relates pressure (P), volume (V), and absolute temperature (T) combined to produce an interesting result. The equation of the state of a perfect gas is as follows : PV = nRT P = Pressure V = Volume n = amount of substance R = ideal gas constant T = Temperature

Common Forms

There are four common forms in which the ideal gas law can be expressed and used in the questions to understand them better. First, from the equation of the state of a perfect gas, let’s look at it.

Most used form:

PV = nRT = nK_bN_aT P = Pressure V = Volume n = number of gas moles R = universal gas constant K_b = Boltzmann constant N_a = Avogadro constant T = absolute Temperature

Molar Form:

As an ideal gas, the number of moles is directly proportional to the number of molecules present in the container. n = m/M m = mass of the gas (in Kgs) M = Molar Mass (in Kgs) PV = mRT/M P = mRT/( MV) The ideal gas law is a functional relationship in thermodynamics that relates to a perfect gas’s pressure, temperature, and density.

Statistical Mechanics:

In Statistical Mechanics, the following information is derived by the ideal gas law for the equation of state of perfect gas: PV = nK_bN₁T Kb= R/N_a P =1VmuM_uK_bT P = Pressure Kb = Boltzmann Constant T = Temperature m = mass of the gas (in Kgs) u = average particle mass M_u = atomic mass constant Thus, this equation comes out when the gas law is transformed into some other forms that are given here.

Combined gas law:

The combined gas law, which takes the same functional form as the ideal gas law, states that the number of moles of gas is unspecified, and the ratio of PV to T is taken as a constant. PV/T = constant P₁V₁/T₁ = P₂V₂/T₂

Conclusion

In the article, we learned about the equation of the state of perfect gas and its dimensional analysis. Apart from it, we learned more about the ideal gas law and how it works in Thermodynamics. Besides this, the perfect gas law and its major types of equation writing were also taught to us.