# Behavior of Real Gases

**Behavior of Real gases (Deviation from ideal behavior) **

**Ideal gas **is that which obeys ideal gas equation **PV = nRT **under all conditions of temperature and pressure.

No gas is ideal. All gases are real and show deviation from ideal gas behavior

**Study of deviation **→ As PV = const. (Boyle’s law) so graph bet PV (v/s) P should be straight line parallel to x-axis. But real gases don’t show such a behavior.

The extent to which a real gas deviates from ideal behavior is studied is terms of compressibility factor ‘Z’ & **Z = PV/nRT**

For ideal gases, PV = nRT & Z = 1

For real gases, PV ≠ nRT so

(i) Of Z < 1 (ep. CH_{4}, CO_{2} etc) the gas shows negative deviation i.e. gas is more compressive than expected from ideal behavior.

(ii) Of Z > 1 (ep. H_{2}, He etc) the gas shows positive deviation i.e. gas is less compressive than expected from ideal behavior.

The curve of Z (v/s) P for N_{2} at different temperatures indicate that **at high temperature **& **low pressure **the value of Z approaches = 1. i.e. behavior ideally

**Boyle temperature **or **Boyle point **→ It’s the temperature at which a real gas behaves like an ideal gas over an appreciable range of pressure.

**Significance of Z **→ Z = (PV_{real}/nRT) (i) of gas shows

Ideal behavior then PV_{ideal} = nRT i.e. V_{ideal} = nRT/P

So from equation (i) **Z = V _{real}/V_{ideal}**

**Cause of deviation from ideal behavior:- **At low temperature and low pressure. The gases behave ideally & at low temperature & high pressure ideal behavior.

(i) The volume occupied by gas molecules is negligible in compassion to total volume of gas.

(ii) The force of attraction & or repulsion between gas molecules is negligible.

**At low T & high P **→ (i) The actual mol of gas is not negligible in compassion to total volume.

(ii) The force of attraction or repulsion is not negligible.

**Equation of state for the real gases **→ **(Vander Waal****’****s equation) **→ (in 1873)

(P + a/V^{2}) (V – b) = RT for 1 mol of gas

(P + an^{2}/V^{2}) (V – nb) = nRT (for n moles of gas)

**Deviation of Vander Waal****’****s equation **→

**Correction for volume **→

Suppose volume of gas molecule = v

Effective volume = b = 4 x v (excluded volume)

So volume of gas = V – b (for 1 mol)

V – nb (for n moles)

**Correction for pressure **→ The molecule colliding with the half of container is attracted by other molecules so it exerts lessee pressure.

So corrected pressure = P + þ

Þ ∝ (density)^{2} but d ∝ (1/V) or d ∝ (n/V)

(1 mol) (n moles)

So Þ ∝ (1/V^{2}) (1 mole)

Þ = a/V^{2}

Þ ∝ (n^{2}/V^{2}) (for n moles)

Þ = (an^{2}/V^{2})

Where a → constant

In Vander Waal’s equation,

(P + an^{2}/V^{2}) (V – nb) = nRT

‘a’, & ‘b’, are Vander Waal’s constant their significance is

‘a’ → Greater the value of ‘a’ for a gas greater is the magnificence of attractive forces between gas molecules. & more easily the gas is liquefiable.

‘b’ → It is a measure of effective size of gas molecule greater the value of ‘b’ higher is the size of gas molecules.

**Units of ****‘****a****’** As P = (an^{2}/V^{2}) or a = (PV^{2}/n^{2}) = atm. L^{2} mol^{-2} or bar dm^{6} mol^{-2}

**Units of ****‘****b****’** As v = nb or b = v/n = L mol^{-1} or dm^{3} mol^{-1}

**Explanation of behavior of real gases by V.W equation**

**(i) At very low pressure **→ V is very large, (a/V^{2}) = small

b → neglected so **PV = RT **the gas behaves ideally.

**(ii) At Moderate pressure **→ V = small (a/V^{2}) = large

V → still larger than b so

(P + a/V^{2}) (V) = RT or PV + (a/V) = RT

Or PV = RT – (a/V)

Or (PV/RT) = 1 – (a/RTV)

Or Z = 1 – (a/RTV) i.e. **Z < 1**

**(iii) At Greater P **→ V → very small, (a/V^{2}) large but P is more

So (a/V^{2}) neglected, but ‘b’ can’t be neglected

So P(V – b) = RT or PV – Pb = RT

Or PV = RT + Pb or (PV/RT) = 1 + Pb i.e. **Z > 1**

Is negligible i.e. a → very = 2 small (a/V^{2}) = negligible

So P(V – b) = RT or Z > 1 & increases with increases is value of P at constant P.