Question

Monochloroethylene is used to make polyvinylchloride (PVC). It has a density of \(2.56 \mathrm{g} / \mathrm{L}\) at \(22.8^{\circ} \mathrm{C}\) and 756 mmHg. What is the molar mass of monochloroethylene? What is the molar volume under these conditions?

Short answer

The molar mass of monochloroethylene is 0.000114 kg/mol, and its molar volume under the given conditions is 22.8 L/mol

Step-by-step solution

Finding the molar mass

Firstly, it is important to understand that the molar mass of a gas can be calculated by dividing its density by its molar volume. Since the molar volume of a gas at tandard conditions (which is approximately close to the conditions stated in this problem: 22.8 degrees Celsius and 756 mmHg) is \(22.4 L/mol\), the molar mass (M) of monochloroethylene can be calculated as follows: \(M = 2.56 g/L ÷ 22.4 L/mol = 0.114 g/mol\)

Convert g/mol to Kg/mol

Scientifically, the molar mass should be in Kg/mol, so the calculated molar mass needs to be converted from g/mol to kg/mol, by dividing it by 1000. Therefore, \(M = 0.114 g/mol ÷ 1000 = 0.000114 kg/mol\)

Calculate the molar volume

Secondly, you have to calculate the molar volume of the gas under the given conditions. To achieve this, use the ideal gas law formula: \(P×V=n×R×T\), where P is the pressure, V is the volume, n is the number of moles, R is the gas constant and T is the temperature in Kelvin. We know n/molar mass = density, and R = 0.0821 L·atm/(K·mol). We convert the pressure from mmHg to atm and temperature from Celsius to Kelvin, and then solve for volume, V. Hence, the molar volume (V) of the gas can be found as follows: \(V= ( nR_{t}T_{1} ) / P_{2} = (density × R × (22.8 + 273.15)) / (756 ÷ 760) = 22.8 L/mol\)

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) represents the number of moles, \(R\) is the proportionality constant known as the ideal gas constant, and \(T\) is the temperature in Kelvin. This law helps us understand how gases will react to changes in temperature, pressure, and volume.

By rearranging this formula, you can solve for different variables, making it a versatile tool. For instance, if we need to find the volume occupied by a gas, knowing the other parameters, we can re-arrange it to \(V = \frac{nRT}{P}\).

• The ideal gas constant \(R\) is always necessary and has a value of 0.0821 L·atm/(K·mol).
• Always remember: Pressure should typically be in atmospheres (atm), and temperature in Kelvin for the calculation to work.

Knowing this law is crucial for solving many problems related to gases, from calculating volumes to understanding moles and how temperature affects a gas.

Density of Gases

Density is another critical concept when dealing with gases. For gases, density \((\rho)\) is defined as mass per unit volume and is often expressed in grams per liter (g/L) in scientific contexts. For a given gas, density can change with temperature and pressure, but it offers a snapshot of the gas's heaviness under specific conditions.

To find the density of a gas under specific conditions, you can employ the relationship: \(\rho = \frac{PM}{RT}\), where \(P\) is pressure, \(M\) is molar mass, \(R\) is the gas constant, and \(T\) is the temperature. This formula links density with the ideal gas law, illustrating that higher molar masses or pressures result in higher densities, while higher temperatures result in lower densities.

• A useful way to remember this: the denser the gas, the greater its molar mass or lower its temperature.
• Density also explains why some gases float, while others sink.

Understanding gas density is not only vital for theoretical problems but is also applied in real-world scenarios such as meteorology and aerospace engineering.

Molar Volume

Molar volume is the volume one mole of gas occupies under specific conditions of pressure and temperature. At standard temperature and pressure (STP — 0°C and 1 atm), the molar volume of an ideal gas is 22.4 L/mol. This standard value is handy because it allows chemists to quickly estimate the volume that gases will occupy.

However, the actual conditions can differ, and that’s why the ideal gas law is useful. By using \(PV = nRT\), we understand how the molar volume shifts when temperature or pressure changes.

• Molar volume changes inversely with pressure — increase the pressure, and the molar volume drops.
• Conversely, it changes directly with temperature — raise the temperature, and the molar volume rises.

When problems provide non-standard conditions, like different temperatures or pressures, molar volume must be calculated accordingly. Calculating molar volume this way helps us comprehend the behavior of gases beyond simple STP scenarios.

Gas Laws

Gas laws are the rules that describe how gases behave, providing essential insights into processes involving gases. Generally, they relate the pressure, temperature, volume, and amount of gas. These principles include Boyle's Law, Charles's Law, Avogadro's Law, and the combined gas law, ultimately forming the basis of the ideal gas law.

• Boyle's Law: At a constant temperature, the volume of a gas is inversely proportional to its pressure. \(PV = k\).
• Charles's Law: At a constant pressure, the volume of a gas is directly proportional to its temperature. \(\frac{V}{T} = k\).
• Avogadro's Law: At a fixed temperature and pressure, the volume of a gas is directly proportional to the number of moles. \(\frac{V}{n} = k\).
• Combined Gas Law: Combines Boyle's and Charles’s laws into one formula: \(\frac{PV}{T} = k\).

These laws help explain why hot air balloons rise or why soda fizzes when opened. Understanding gas laws is invaluable in predicting how changes affect the behavior of gases.