# Combined Gas Law or the Gas Equation

**Combined gas law**states that, for a given mass of a gas, the product of its pressure and volume divided by its Kelvin temperature is**a constant**.**Boyle’s Law**and**Charle’s Law**give separately the effect of pressure and temperature respectively on the volume of a gas.**Combined Gas Law**or the**gas equation**gives the simultaneous effect of the changes in pressure and temperature on the volume of a gas.- It can be derived by
**combining Boyle’s Law and Charle’s Law**as follows:

**Let us** suppose a certain quantity of a gas in initial state has **volume V _{1}** at temperature

**T**and

_{1}**pressure P**. Let in

_{1}**final state**the temperature and pressure be changed to

**T**respectively, so that the volume changes to

_{2}and P_{2}**V**

_{2}.**Further, suppose** that this change is brought about in **two steps** as shown in figure. **In the first step,** suppose temperature is kept constant at **T _{1}**, then, if the pressure is changed from

**P**suppose volume changes

_{1}to P_{2},**from V**.

_{1}to V*Image source: ck12*

**Applying Boyle’s Law**, we have

**P _{1}V_{1} = P_{2}V**

**V = P _{1}V_{1}/P_{2}**……………………………………… (i)

**In the second step**, suppose pressure is kept constant at **P _{2}**. Now, if the temperature is changed from

**T**, the volume changes from

_{1}to T_{2}**V to V**.

_{2}

**Applying Charle’s Law**, we have

**V/T _{1} = V_{2}/T_{2}**…………………………………… (ii)

Substituting the value of **V** from equation (i) in equation (ii), we get

**P _{1}V_{1}/P_{2}T_{1}= V_{2}/T_{2}**

Or**, P _{1}V_{1}/T_{1} = P_{2}V_{2}/T_{2}**………………….. (iii)

This implies that,

**PV/T** = a constant (**say k**)

Or**, PV = kT**………………………………………. (iv) where** k** is proportionality constant

**Numerical value** of proportionality constant depends upon the **quantity of gas** and the units in which volume and pressure are expressed but is **totally independent** of the nature of the gas.

**For 1 mole of any gas**, the constant **‘k’** is termed as the **universal gas constant** and is denoted by **R**

**PV = RT**………………………………………………. (v)

**Equation (v)** is known as **the general gas equation** or **the ideal gas equation** for one mole of gas.

**For ‘n’ moles of a gas**, therefore the gas equation may be written as

**PV = n R T**……………………………………………… (vi)

*Image source: bartleby * *Image source: gaslaws*

**Nature of Gas Constant (R)**

The exact nature of ‘R’ can be understood by considering the quantities involved in the gas equation.

From gas equation, *PV = n R T,* we have

*R = PV/nT* = pressure × volume/(number of moles × absolute temperature)

= force/area × volume/(no. of moles × absolute temperature)

= force/(length^{2}) × (length^{3})/ (no. of moles × absolute temperature)

= (force × length)/(no. of moles ×absolute temperature)

= work/(no. of moles × absolute temperature)

Thus**,’R’** represents work done per degree per mole. In CGS units it is expressed as **ergs K ^{-1} mole^{-1}** and in SI units as

**Joule K**.

^{-1}mole^{-1}**References: **

i) https://tyrocity.com/chemistry-notes/combined-gas-equation-3o36

ii) https://www.chem.fsu.edu/chemlab/chm1045/gas_laws.html