Chapter 1, Problem 88A

To determine

The density of a gas for a given pressure .

Answer to Problem 88A

The density of the gas for the given pressure is $49.38{\text{ kg/m}}^3$.

Explanation of Solution

Given:

A container of gas with a pressure of 101 kPa has a volume of 324 cm³ and mass of 4.00 g. The pressure is increased to 404 kPa.

Formula Used:

The density of a substance ratio of its mass to unit volume. It is mathematically expressed as,

  $\rho =\frac{m}{V}$

Since mass is a constant, the above equation can be rewritten as,

  $\rho \propto \frac{1}{V}$.......(1)

The pressure of a substance is inversely proportional to the volume and this can be expressed as,

  $P\propto \frac{1}{V}$.......(2)

The equations (1) and (2) implies

  $\rho \propto P$

Calculation:

The gas has a pressure of 101 kPa has a volume of 324 cm³ and mass of 4.00 g. The density of the gas is obtained as,

  $\begin{array}{c}\rho =\frac{4\times {10}^{-3}}{324\times {10}^{-6}}\\ =12.35{\text{ kg/m}}^3\end{array}$

The pressure is now increased to 404 kPa. This is four times the initial value, and the relation between density of the gas and its pressure implies an increase in density by the same factor. Thus, the new density of the gas is given by,

  $\begin{array}{c}{\rho }_{new}=4\rho \\ =4\times 12.35{\text{ kg/m}}^3\\ =49.38{\text{ kg/m}}^3\end{array}$

Conclusion:

The density of a gas for a given pressure is $49.38{\text{ kg/m}}^3$ .