Problem 29 - General Chemistry: Principles and Modern Applications

Question

\(\mathrm{Kr}(\mathrm{g})\) in a 18.5 L cylinder exerts a pressure of \(11.2 \mathrm{atm}\) at \(28.2^{\circ} \mathrm{C} .\) How many grams of gas are present?

Step-by-Step Solution

Answer

The mass of the Krypton gas in the cylinder is 73.24 grams.

Step 1: Identify Given Values

The values given in the problem are Pressure(P) = 11.2 atm, Volume(V) = 18.5 L, Temperature(T) = \(28.2^{\circ} \mathrm{C}\) (which needs to be converted to Kelvin by adding 273.15), and the gas constant(R) = 0.0821 L.atm/K.mol.

Step 2: Convert Temperature to Kelvin

The temperature is given in degrees Celsius but the ideal gas law requires temperature to be in Kelvin. Therefore, convert the temperature to Kelvin by adding 273.15 to the Celsius temperature. This gives \(T = 28.2 + 273.15 = 301.35 \mathrm{K}\).

Step 3: Substitute values into the Ideal gas law

Substitute the values of the pressure, volume, gas constant, and temperature into the ideal gas law (\(PV = nRT\)). This gives \(11.2 \mathrm{atm} * 18.5 \mathrm{L} = n * 0.0821 * 301.35 \mathrm{K}\). Solve for n.

Step 4: Solve for n

Solve the equation for n, the number of moles. \( n = \frac{11.2 \mathrm{atm} * 18.5 \mathrm{L}}{0.0821 * 301.35 \mathrm{K}} = 0.873 \mathrm{moles}\).

Step 5: Convert moles to grams

Convert the number of moles to grams using the molar mass of Krypton, which is 83.8 g/mol. This gives \(\mathrm{Mass} = 0.873 \mathrm{moles} * 83.8 \mathrm{g/mol} = 73.24 \mathrm{g}\).

Key Concepts

Krypton gas calculations

When dealing with gases under varying conditions, the Ideal Gas Law is a fundamental concept. This law can help us understand and find particular properties of a gas's state under specific temperature, pressure, and volume. The Ideal Gas Law equation is written as:

• \( PV = nRT \) where:
  • \( P \) is the pressure (in atmospheres, atm)
  • \( V \) is the volume (in liters, L)
  • \( n \) is the number of moles of the gas
  • \( R \) is the ideal gas constant \( (0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1}) \)
  • \( T \) is the temperature (in Kelvin, K)

When calculating the amount of a specific type of gas, like Krypton, it becomes crucial to correctly apply these variables and solve for the unknown. In our case, we had a known pressure, volume, and temperature, and we used the Ideal Gas Law to solve for \( n \), the number of moles.Once \( n \) is determined from the formula, this value is then employed to find other properties of the gas, such as its mass, as we'll see next.

Mole-to-gram conversion

After finding the number of moles of Krypton gas using the Ideal Gas Law, the next step is to convert moles into grams. The ability to switch between moles and grams is a handy skill in chemistry. This requires using the molar mass of the substance, which for Krypton, is 83.8 grams per mole.Here's how you can convert moles to grams:

• Identify the molar mass of the substance. For Krypton, it's \(83.8\, \text{g/mol}\).
• Use the formula: \[ \text{Mass (g)} = \text{Number of moles} \times \text{Molar mass (g/mol)} \]
• In the exercise, this calculation was: \[ \text{Mass} = 0.873\, \text{moles} \times 83.8\, \text{g/mol} = 73.24\, \text{g} \]

This means that there are 73.24 grams of Krypton in the cylinder, providing a tangible understanding by converting moles into a more relatable mass measurement.

Temperature conversion to Kelvin

When using the Ideal Gas Law, it's essential to have the temperature in Kelvin. Kelvin is the absolute temperature scale in scientific measurements and ensures accurate calculations. Here's how you convert from Celsius to Kelvin:

• Start with the temperature in Celsius. In our problem, it was \(28.2^{\circ} \text{C}\).
• Add 273.15 to the Celsius temperature to convert directly to Kelvin. This gives us \(301.35\, \text{K}\).

Kelvin does not have degrees because it starts from absolute zero, the lowest possible temperature, a concept rooted in thermodynamics. Ensuring the temperature is in Kelvin guarantees that all variables in the Ideal Gas Law fit the standard units required for accurate results.