Question

Using the equation of state pV = nRT; show that at a given temperature density of a gas is proportional to gas pressure p.

Answer 1

The equation of state is given by,

pV = nRT ……….. (i)

Where,

p → Pressure of gas

V → Volume of gas

n→ Number of moles of gas

R → Gas constant

T → Temperature of gas

From equation (i) we have,

`"n"/"V" = "p"/"RT"`

Replacing n with `m/M` we have

`"m"/("MV") = "p"/"RT"`   ...(ii)

Where,

m → Mass of gas

M → Molar mass of gas

But `m/V = d` (d = density of gas)

Thus, from equation (ii), we have

`"d"/"M" =  "p"/"RT"`

`=> "d" = ("M"/"RT")"p"`

Molar mass (M) of a gas is always constant and therefore, at constant temperature (T), `"M"/"RT"`= constant.

`"d"  = ("constant") " p"`

`=> "d" prop "p"`

Hence, at a given temperature, the density (d) of gas is proportional to its pressure (p).