A cylinder contains nitrogen gas. A piston compresses the gas to half its initial volume. Afterward, has the mass density of the gas changed? If so, by what factor? If not, why not?

Verified step by step guidance

1. Step 1: Understand the relationship between mass density and volume. Mass density (ρ) is defined as the mass (m) of the gas divided by its volume (V): $\rho =\frac{m}{V}$. The mass of the gas remains constant during compression, but the volume changes.
2. Step 2: Analyze the effect of halving the volume. If the volume is reduced to half its initial value, the denominator in the density formula decreases, causing the density to increase.
3. Step 3: Determine the factor by which the density changes. Since the volume is halved, the density will double. Mathematically, if the initial density is $\rho$₀, the new density becomes $\rho$_new=2ρ₀.
4. Step 4: Confirm that the mass remains constant. The mass of the gas does not change during compression because no gas is added or removed from the cylinder.
5. Step 5: Conclude that the mass density has changed and increased by a factor of 2 due to the halving of the volume.

Key Concepts

Mass Density

Mass density is defined as the mass of a substance per unit volume, typically expressed in kilograms per cubic meter (kg/m³). It is a crucial property that indicates how much mass is contained in a given volume of material. In the context of gases, density can change with variations in pressure and temperature, but remains constant if the mass of the gas does not change.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law helps in understanding how changes in volume and pressure affect the behavior of gases, particularly during compression.

Conservation of Mass

The principle of conservation of mass states that mass cannot be created or destroyed in a closed system. In the context of the gas in the cylinder, compressing the gas to half its initial volume does not change the total mass of the gas present. Therefore, while the volume decreases, the mass density increases proportionally, as the same mass is now occupying a smaller volume.