Problem 142 - General Chemistry: Principles and Modern Applications

Question

To establish a pressure of 2.00 atm in a 2.24 L cylinder containing \(1.60 \mathrm{g} \mathrm{O}_{2}(\mathrm{g})\) at \(0^{\circ} \mathrm{C},\) (a) add \(1.60 \mathrm{g} \mathrm{O}_{2} ;(\mathrm{b})\) add \(0.60 \mathrm{g} \mathrm{He}(\mathrm{g}) ;(\mathrm{c})\) add \(2.00 \mathrm{g} \mathrm{He}(\mathrm{g})\) (d) release \(0.80 \mathrm{g} \mathrm{O}_{2}(\mathrm{g})\)

Step-by-Step Solution

Answer

Each action should be performed as follows: (a) Add 1.60 g of O2. (b) Add 0.694 g of He. (c) Add 2.00 g of He. (d) Release 0.846 g of O2.

Step 1: Convert temperature to Kelvin

Temperature in the Ideal Gas Law must be in Kelvin. Convert 0 degrees Celsius to Kelvin using the formula K = °C + 273.15. Hence, the temperature is 273.15 K.

Step 2: Calculate number of moles to achieve desired pressure

Rearrange the Ideal Gas Law to solve for n, the number of moles. n=PV/RT. Substitute P=2.00 atm, V=2.24 L, R=0.0821 L.atm/(mol.K) (Ideal Gas Law constant), T=273.15 K into the formula. This gives the required moles of gas.

Step 3: Calculate mass for given substances

The number of moles obtained in step 2 is used to compute the mass of each gas using their molar masses. (a) For O2 the molar mass is 32.00 g/mol and for He it's 4.00 g/mol. (b, c) Compute the masses for 0.60g of He and 2.00g of He. (d) Deduct mass of 0.8g O2 from the initial 1.6g O2.

Step 4: Add or release required amount of each gas

Add or release the required amount of each gas found in step 3 to the cylinder to achieve the pressure of 2.00 atm.

Key Concepts

Moles Calculation

Understanding the concept of moles is crucial when dealing with gases and their reactions. The mole is a unit that represents a specific number of particles, typically atoms or molecules. In chemistry, the number of moles helps understand quantities at the atomic scale. You can calculate the number of moles using the Ideal Gas Law, represented as \( n = \frac{PV}{RT} \). Here:

• \( n \) is the number of moles of the gas.
• \( P \) represents the pressure in atmospheres.
• \( V \) is the volume in Liters.
• \( R \) is the Ideal Gas Constant (\(0.0821\, \text{L.atm/(mol.K)}\)).
• \( T \) is the temperature in Kelvin.

By rearranging the formula, you can solve for \( n \) using the known variables such as pressure, volume, and temperature. This calculation is pivotal for understanding how much gas is needed or produced in a given reaction.

Gas Pressure

Gas pressure is the force that gas molecules exert per unit area when they collide with the walls of their container. It is measured in different units including atmospheres (atm). The Ideal Gas Law, \( PV = nRT \), connects pressure with other factors like number of moles, volume, and temperature. An increase in the number of gas molecules in a constant volume results in increased pressure because more particles are colliding with the container walls.It's also interesting to note how the addition or release of gas can affect pressure. For example, adding helium to an existing oxygen gas will increase the total pressure. This concept is crucial when adjusting the gas until you reach a desired pressure like the 2.00 atm required in the cylinder example.

Temperature Conversion

In the context of gas laws, it is vital to work with the absolute temperature scale, which is Kelvin. The Kelvin scale is directly connected to the energy of particles, as it starts from absolute zero - a theoretical point where particle motion ceases.Converting Celsius to Kelvin is straightforward. Simply add 273.15 to the Celsius measurement. For example, \(0^\circ \text{C}\) converts to \(273.15\, \text{K}\).This step is essential as the Ideal Gas Law requires temperature to be in Kelvin, allowing for accurate calculations of volume, pressure, and number of moles. Thus, always remember:

• Celsius to Kelvin: K = °C + 273.15

Molar Mass

The molar mass of a substance is the mass of one mole of its molecules or atoms, expressed in grams per mole (g/mol). It allows us to convert between the mass of a substance and the number of moles. For gases, the molar mass plays a crucial role in understanding how to achieve certain pressures with given amounts. Different elements have different molar masses, for example:

• Oxygen (\(O_2\)) has a molar mass of \(32.00\, \text{g/mol}\).
• Helium (He) has a molar mass of \(4.00\, \text{g/mol}\).

When you know the mass of your gas, you can calculate the number of moles by dividing the mass by the molar mass. This is a key step to determine how much you need to add or remove to reach a specific state, such as achieving the desired 2.00 atm pressure in the exercise example.