Question

Establish the equation

P V = R T

for an ideal gas and obtain the dimensional formula for it

R

.

Accepted Answer

The study of gases shows that the volume, pressure and temperature of a given mass of gas are interrelated to one another. If we fix the values of any two of them, then the value of the third cannot be changed.

However, there is a relation between the pressure, volume and temperature of a given mass of a gas. This relation is called ‘the perfect gas equation’.

Suppose, initially the pressure, volume and absolute temperature of a given mass of gas are $P_{1}$ , $V_{1}$ and $T_{1}$ respectively and they finally change to $P_{2}$ , $V_{2}$ and $T_{2}$ respectively. We can divide this change into two parts:

$(1)$ Suppose the temperature of the gas is kept constant at $T_{1}$ and the pressure is changed from $P_{1}$ to $P_{2}$ . If in this process the volume changes from $V_{1}$ to $V^{'}$ then by Boyle’s law, we have $P_{2} V^{'} = P_{1} V_{1}$ or

$V^{'} = \frac{P_{1} V_{1}}{P_{2}} . . . . . (i)$

$(2)$ Now, suppose the pressure of the gas is kept constant at $P_{2}$ and its absolute temperature is changed from $T_{1}$ to $T_{2}$ . If in this process the volume of the gas changes from $V^{'}$ to $V_{2}$ , then by Charles’ law, we have,

$\frac{V^{'}}{T_{1}} = \frac{V_{2}}{T_{2}}$ or

$V^{'} = \frac{T_{1} V_{2}}{T_{2}} . . . . . (i i)$

Comparing the two equations,

$\frac{P_{1} V_{1}}{T_{1}} = \frac{P_{2} V_{2}}{T_{2}}$ or

$\frac{P V}{T} = r$ (constant) or

$P V = r T$

This is the perfect gas equation. The value of the gas constant $r$ depends upon the nature and mass of the gas. Its value is different for different masses of the same gas or for different gases of the same mass. It is also called a specific gas constant.

However, if we take $1$ gram-molecule (1 mole) of gas, the value of gas-constant will be the same for all gases. Then it is called ‘universal gas constant’ and is represented by $R$ . Hence, for $1$ mole of gas, the gas equation takes the following form:

$P V = R T$ .

Dimensions of $R = [M L^{2} T^{- 2} θ^{- 1}]$

G L Mittal and TARUN MITTAL Solutions for Chapter: Kinetic Theory of Gases, Exercise 1: QUESTIONS

G L Mittal Physics Solutions for Exercise - G L Mittal and TARUN MITTAL Solutions for Chapter: Kinetic Theory of Gases, Exercise 1: QUESTIONS

Questions from G L Mittal and TARUN MITTAL Solutions for Chapter: Kinetic Theory of Gases, Exercise 1: QUESTIONS with Hints & Solutions

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