## Combined Gas Laws:

To arrive at a combined relationship between pressure, volume and temperature of a given mass of a gas, Boyle’s and Charle’s law are combined into one equation. The equation so obtained is called the Gas Equation or the Equation of State.

Let P_{1}, V_{1}, T_{1} be the original (initial) pressure, volume and temperature of a gas, respectively. Let the Pressure P_{1} change to P_{2}, temperature T_{1} to T_{2} in two successive stages as the volume of gas changes from V_{1} to V_{2}.

In the first stage, let us suppose that the volume changes from V_{1} to V (intermediate volume) when there is a change in pressure from P_{1} to P_{2} at constant temperature T_{1}.

According to Boyle’s Law- P _{1}V_{1} = P_{2}Vor V = P _{1}V_{1}/P_{2} ……………..(I) |

In the second stage, when the pressure remains constant at P_{2}, consider the change of temperature from T_{1} to T_{2} producing a change in volume from V to V_{2}.

According to Charle’s Law- V/T _{1} = V_{2}/T_{2}or V = V _{2}T_{1}/T_{2} ……………………(II) |

Equating V from (I) and (II), we get-

P_{1}V_{1}/P_{2} = V_{2}T_{1}/T_{2}or P _{1}V_{1}/T_{1} = P_{2}V_{2}/T_{2} ……………………(III)or PV/T = a constant ……………………(IV) |

Equation (III), called the **gas equation**, is very useful for the conversion of gas volume from one temperature and pressure to another.

The equation (IV) can be generalized in light of **Avogadro’s law** according to which the volume occupied by one gram molecule of every gas is the same under similar conditions of temperature and pressure.

Thus, for one gram molecule (i.e., 1 mole) of every gas, the expression PV/T must have the same value and the later is represented by **R**. ‘R’ is known as the **molar gas constant**. We may write now,

PV/T = R (for 1 mole of a gas) or PV = RT (Gas Equation) For ‘n’ moles of a gas, the gas equation is written as- PV =nRT |

## Numerical Value of R:

The numerical value of R is calculated by making use of the expression PV/T = R and taking an advantage of the fact that the volume occupied by 1 gram molecule of any gas at 0°C and 1 atmospheric pressure is 22.4 litres. It varies with the units in which pressure and volume are expressed.

- When the pressure is taken in atmospheres and the volume in litres-

P = 1 atmosphere; V = 22.4 litres; T = 273 K R = PV/T = 1 x 22.4/273 = 0.0821 litre atmopshere per Kelvin per mole |

- In
**C.G.S. Units**–

P = 76 x 13.6 x 981 dynes/sq. cm; V = 22,400 cc; T = 273 K R = PV/T = 76 x 13.6 x 981 x 22400/273 = 8.314 x 10^{7} ergs per Kelvin per mole |

- In
**S.I. Units**–

P = 1.0133 x 10^{5} N/m^{2}; V = 0.022414 m^{3}; T = 273 KR = PV/T = 1.0133 x 10 ^{5} x 0.022414/273 = 8.314 Joules per Kelvin per mole |

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