Using the ideal gas equation (pv=nRT)show that at a given temperature density of a gas is proportional to gas pressure p

The ideal gas equation is given by:

$$PV = nRT$$

where:
$P$ = pressure
$V$ = volume
$n$ = number of moles
$R$ = ideal gas constant
$T$ = temperature

The number of moles ($n$) of a gas is its mass ($m$) divided by its molar mass ($M$):

$$n = \frac{m}{M}$$

Divide both sides by $V$:

$$P = \left(\frac{m}{V}\right)\frac{RT}{M}$$

Recognize that $\frac{m}{V}$ is the density ($\rho$):

$$P = \rho \frac{RT}{M}$$

For a given gas (meaning $M$ is constant) and at a given temperature ($T$ is constant), the terms $\frac{M}{RT}$ are all constant. Let's call this constant $k$:

$$k = \frac{M}{RT}$$

Therefore, the equation becomes:

$$\rho = Pk$$

This shows that density ($\rho$) is directly proportional to pressure ($P$) when the temperature and molar mass are constant.

Thus, at a given temperature, the density of a gas is proportional to the gas pressure $P$.