Chapter 2: Problem 199

Explain what effect compressing a gas has on the density of the gas.

Short Answer

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Compressing a gas leads to an increase in its density because the pressure inside the gas increases while the temperature remains constant. This relationship can be observed through the equation \(\rho = \frac{MP}{RT}\), where \(\rho\) is the density, M is the molecular weight, P is the pressure, R is the gas constant, and T is the temperature. Since density is directly proportional to pressure and inversely proportional to temperature, an increase in pressure due to compression and constant temperature results in increased density.

Step by step solution

Understand the Ideal Gas Law

The ideal gas law is an equation of state for a hypothetical ideal gas. It relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the number of molecules (n) and the gas constant (R). The ideal gas law is given by the equation: \[PV = nRT\] Where, P = Pressure of the gas V = Volume of the gas n = Number of molecules of the gas R = Gas constant (8.314 J/(mol·K)) T = Temperature of the gas in Kelvin

Define Density and Relate it to Gas Laws

Density (\(\rho\)) is defined as mass per unit volume: \[\rho = \frac{m}{V}\] Now, we know that the number of molecules (n) in a gas is related to its mass (m) by the molecular weight (M) of the gas: \[n = \frac{m}{M}\] We can rewrite the ideal gas law equation using density: \[P = \frac{nRT}{V} = \frac{mRT}{MV}\] Now, we replace the mass-to-volume ratio with density and get the equation relating the density of a gas to its pressure and temperature: \[\rho = \frac{MP}{RT}\]

Discuss the Effect of Compression on Gas Density

Compressing a gas means decreasing its volume while keeping the temperature constant. When we decrease the volume of the gas, the pressure inside the gas increases in order to maintain the ideal gas law relationship (since temperature is constant). According to the density equation we derived above, the density of a gas (\(\rho\)) is directly proportional to the pressure (P) and inversely proportional to the temperature (T). Thus, when the pressure increases due to compression and temperature remains constant, the density of the gas also increases. In conclusion, compressing a gas increases its density because the pressure inside the gas increases while the temperature remains constant.

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Explain what effect compressing a gas has on the density of the gas.

Short Answer

Step by step solution

Understand the Ideal Gas Law

Define Density and Relate it to Gas Laws

Discuss the Effect of Compression on Gas Density

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