Question

A cylinder contains nitrogen gas.A piston compresses the gas

to half its initial volume.

a. Has the mass density of the gas changed? If so, by what factor?

If not, why not?

b. Has the number of moles of gas changed? If so, by what factor?

If not, why not?

Short answer

(a) The density goes twice of its initial stage.

(b) The number of moles doesn't change.

Step-by-step solution

Change in the mass density 

To describe the microscopic system, the state variables are used to describe the state of the system. One of the state variables is density$\rho$and is defined as the ratio of mass and volume. In equation form, it is

localid="1648473586903" $\rho =\frac{M}{V}$

According to equation (1), the densitylocalid="1648473593095" $\rho$and the volume of the gas are in inverse proportion

localid="1648473598671" $\rho \propto \frac{1}{V}$

So, when the gas is compressed, the volume will decrease and the density will increase.

When the final volume is half of the initial volume (localid="1648473604024" $V_2=\frac{1}{2}{V}_1$), the density will be

localid="1648473610720" $$\begin{gathered} \frac{{\rho }_2}{{\rho }_1}=\frac{V_1}{V_2} \\ {\rho }_2=\left(\frac{V_1}{V_2}\right){\rho }_1 \\ {\rho }_2=\left(\frac{V_1}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{V}_1}\right){\rho }_1 \\ {\rho }_2=2{\rho }_1 \end{gathered}$$

So, the final density is twice of its initial density.

Ster 2: Change in the number of moles of gas 

The mass$M$of the gas is related to the number of moles$n$of the gas by

$M=n{M}_{mol}$

Where,$M_{mol}$=the molar mass. When the gas is compressed, its volume and density will change and the mass will be constant. As the mass isn't changing here, the number of moles will not change.