Question

A sample of chlorine gas occupies \(1550 \mathrm{~mL}\) at \(0.945 \mathrm{~atm}\) and \(50^{\circ} \mathrm{C}\). What is the mass of the sample?

Short answer

The mass of the chlorine gas is approximately 3.98 grams.

Step-by-step solution

Understand the Problem

We need to determine the mass of chlorine gas given its volume, pressure, and temperature. We can use the ideal gas law, which relates these quantities to find the number of moles, and then convert moles to mass using chlorine's molar mass.

Convert Temperature to Kelvin

The ideal gas law requires the temperature to be in Kelvin. The conversion from Celsius to Kelvin is done using the formula \[ T(K) = T(°C) + 273.15 \] Substituting the given temperature, we have:\[ T(K) = 50 + 273.15 = 323.15 \text{ K} \]

Use Ideal Gas Law

The ideal gas law is given by \[ PV = nRT \] where \(P\) is the pressure in atmospheres, \(V\) is the volume in liters, \(n\) is the number of moles, \(R\) is the ideal gas constant \(0.0821 \text{ L atm} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}\), and \(T\) is the temperature in Kelvin. First, convert the volume from mL to liters: \[ V = 1550 \text{ mL} = 1.550 \text{ L} \] Then, rearrange the ideal gas law to solve for \(n\):\[ n = \frac{PV}{RT} \]

Calculate the Number of Moles of Chlorine

Substitute the values into the rearranged ideal gas law:\[ n = \frac{(0.945 \text{ atm})(1.550 \text{ L})}{(0.0821 \text{ L atm} \cdot \text{mol}^{-1} \cdot \text{K}^{-1})(323.15 \text{ K})} \]Solving this gives:\[ n \approx 0.0561 \text{ moles} \]

Calculate the Mass of Chlorine

The molar mass of chlorine gas (\(Cl_2\)) is approximately \(70.9 \text{ g/mol}\). Use the number of moles to find the mass:\[ \text{Mass} = n \times \text{molar mass} = 0.0561 \text{ moles} \times 70.9 \text{ g/mol} \]This calculation gives:\[ \text{Mass} \approx 3.9789 \text{ grams} \]

Conclusion

The mass of the chlorine gas sample is approximately \(3.98 \text{ grams}\).

Key concepts

Chlorine Gas

Chlorine gas (\(Cl_2\)) is a greenish-yellow gas with a distinct smell, commonly used in industrial and chemical processes. It's a diatomic molecule, meaning it's composed of two chlorine atoms. Thus, it's represented as \(Cl_2\). In the context of chemistry exercises, chlorine gas can frequently appear when we're calculating the reactive behavior of gases. It has a molar mass of approximately 70.9 g/mol, which we'll use when converting moles to grams.

Chlorine is substantial in studies due to its applications in water purification and disinfection processes. Understanding its properties and calculations is crucial in laboratory scenarios and industrial applications.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). For chlorine gas, which is \(Cl_2\), you calculate the molar mass by adding the atomic masses of two chlorine atoms. Each chlorine atom has an atomic mass of approximately 35.45 g/mol.

Therefore, the molar mass of chlorine gas can be calculated as follows:

• Molar Mass of \(Cl_2 = 35.45 ext{ g/mol} \times 2 = 70.9 ext{ g/mol}\)

This value is crucial for converting moles, derived from the ideal gas equation, into grams. Being able to determine molar mass accurately is essential for solving chemistry problems involving gas behavior and chemical reactions.

Gas Volume Conversion

When working with gases, it's vital to pay attention to the units you use. Volume is frequently given in milliliters (mL) but must be converted to liters (L) for calculations with the ideal gas law. The conversion is straightforward, with 1 liter equaling 1000 milliliters.

• For example, \(1550 \text{ mL} = 1550 \div 1000 = 1.550 \text{ L}\).

This conversion is necessary because the ideal gas constant \(R = 0.0821 \text{ L atm/mol K}\) uses liters in its units. Remembering to convert to the correct units ensures accurate calculations in your exercises involving the ideal gas law.

Temperature Conversion in Chemistry

In chemistry, especially when applying the ideal gas law, temperatures must be in Kelvin. The Kelvin scale is an absolute temperature scale based on absolute zero, unlike Celsius or Fahrenheit. To convert Celsius to Kelvin, you add 273.15 to your Celsius measurement.

• For instance, a temperature of \(50^{\circ}\text{C}\) would convert to Kelvin as follows:
  \[ T(K) = 50 + 273.15 = 323.15 \text{ K} \]

Ensuring temperature is in Kelvin is crucial because the ideal gas law \(PV = nRT\) dictates this requirement. The specifics of gas behavior depend significantly on the correct temperature input, ensuring the formula's functionality and precision becomes optimized.