Question

A cylinder containing 151 lb \(\mathrm{Cl}_{2}\) has an inside diameter of 10 in. and a height of 45 in. The gas pressure is 100 psi \((1 \mathrm{atm}=14.7 \mathrm{psi})\) at \(20^{\circ} \mathrm{C} . \mathrm{Cl}_{2}\) melts at \(-103^{\circ} \mathrm{C},\) boils at \(-35^{\circ} \mathrm{C},\) and has its critical point at \(144^{\circ} \mathrm{C}\) and \(76 \mathrm{atm} .\) In what state \((\mathrm{s})\) of matter does the \(\mathrm{Cl}_{2}\) exist in the cylinder?

Short answer

The state of the Chlorine (Cl2) in the cylinder is gas.

Step-by-step solution

Convert pressure units

The pressure is given in psi, and we need to convert it to atm. Using the conversion rate given (1 atm = 14.7 psi), we can find the pressure in atm. We do this by dividing the given pressure by the conversion rate: \(100 \, \text{psi} \, / \, 14.7 \, \text{psi/atm} = 6.8 \, \text{atm}\)

Compare condition with critical point

Compare the obtained pressure and given temperature with the critical point data. The critical point is given as 144°C and 76 atm. The condition inside the cylinder which is 6.8 atm and 20°C is well below these values.

Determine the state of Cl2

Considering that the pressure and temperature in the tank are below the critical point values, refer to the boiling and melting points of Cl2. The boiling point is -35°C and the melting point is -103°C. Given the temperature inside the tank is 20°C, which is higher than the boiling point, the chlorine in the tank will be in a gaseous state.

Key concepts

Critical Point

The critical point of a substance is a very unique property. It is the exact set of conditions under which the liquid and gas phases of the substance coexist in equilibrium. Beyond this point, there is no distinct phase boundary between liquid and gas. For each substance, the critical point is specified by a critical temperature and a critical pressure.

When you look at chlorine (\(\text{Cl}_2\)), its critical point indicates the highest temperature and pressure at which it can exist as a liquid. For chlorine, this is 144°C and 76 atm. If the temperature and pressure surpass these values, it cannot be liquefied regardless of how much pressure is applied.

Understanding the critical point helps in determining the state of a substance in various environmental conditions, as it gives a reference of the limit beyond which the gas and liquid phases merge into a supercritical fluid.

Pressure Conversion

Pressure conversion is a crucial skill when working with states of matter. Often, pressure may be given in one unit, such as psi (pounds per square inch), but needs to be used or compared with another unit like atm (atmospheres).

To convert pressure from psi to atm, you can use the given conversion factor: 1 atm = 14.7 psi. Simply divide the pressure in psi by this factor to obtain the pressure in atm.

• For example, if you have a pressure of 100 psi, the calculation to convert this to atm is \(\frac{100\, \text{psi}}{14.7\, \text{psi/atm}} = 6.8\, \text{atm}\).

This calculation is essential not only for theoretical exercises but also in practical applications where different systems of units are used. Accurate pressure conversion ensures that comparisons and calculations are precise, which is particularly important when dealing with gases and their properties.

Boiling and Melting Points

Boiling and melting points are fundamental properties that define the physical state of a substance at a given temperature.

The melting point is the temperature at which a solid turns into a liquid. For chlorine (\(\text{Cl}_2\)), the melting point is -103°C. This means that chlorine will be a solid at temperatures below -103°C.

The boiling point, on the other hand, is the temperature at which a liquid becomes a gas. Chlorine boils at -35°C, which means it transitions from liquid to gaseous form at this temperature.

• If the temperature is above the boiling point, chlorine will exist as a gas.
• If it is below the melting point, it will be a solid.

In the context of the original exercise, knowing these points is crucial. Since the temperature inside the tank is 20°C, which is higher than the boiling point (-35°C) but well above the melting point, we can deduce that it is in a gaseous state. Understanding the boiling and melting points helps one predict the state of a substance under different conditions.