#### Question

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

#### Solution 1

The equation of state is given by,

*pV* = *n*R*T* ……….. (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)*

#### Solution 2

According to ideal gas equation

PV = nRT or PV=nRT/V

`n = "Constant Mass of gas"/"Molar mass of gas"`

`P = (mRT)/"MV"` [`:. rho("density") = m/V`]

`P = (rhoRT)/M`

`Pxx rho` at constant temperature