Question

The steel reaction vessel of a bomb calorimeter, which has a volume of \(75.0 \mathrm{~mL}\), is charged with oxygen gas to a pressure of \(14.5\) atm at \(22^{\circ} \mathrm{C}\). Calculate the moles of oxygen in the reaction vessel.

Short answer

There are approximately 0.0396 moles of oxygen in the reaction vessel.

Step-by-step solution

Convert the given values to the appropriate units

We need to convert the given values to the appropriate units to use in the Ideal Gas Law formula. The volume should be in liters, the pressure should be in atm, and the temperature should be in Kelvin. 1. Convert the volume from mL to L: \[75.0\: mL \times \frac{1\:L}{1000\:mL} = 0.075\: L\] 2. The pressure is already given in atm, so we don't need to convert it. 3. Convert the temperature from Celsius to Kelvin: \[22^{\circ}\:C + 273.15 = 295.15\:K\] Now, we have the volume as 0.075 L, the pressure as 14.5 atm, and the temperature as 295.15 K.

Solve for the number of moles using the Ideal Gas Law

Using the Ideal Gas Law formula \(PV = nRT\), we can solve for the number of moles (n) of oxygen gas in the container with the given values: We are given: \(P = 14.5\: atm, V = 0.075\: L, T = 295.15\:K\) and \(R = 0.0821\: \frac{L\:atm}{mol\:K}\). Now we can rearrange the formula to solve for n: \[n = \frac{PV}{RT}\] Substitute the given values into the equation: \[n = \frac{(14.5\:atm)(0.075\:L)}{(0.0821\: \frac{L\:atm}{mol\:K})(295.15\:K)}\]

Calculate the number of moles of oxygen

Now, we can do the calculations to find the number of moles of oxygen gas in the container: \[n = \frac{(14.5)(0.075)}{(0.0821)(295.15)} = 0.0396\:mol\] Hence, there are approximately 0.0396 moles of oxygen in the container.

Key concepts

Moles of Gas

Understanding moles is key when working with gas calculations. Moles (mol) are used to measure the amount of a substance. In chemistry, it's a standard unit that makes it easier to express chemical amounts and reactions.
The mole concept is based on Avogadro's number, which is approximately \(6.022 \times 10^{23}\). This is the number of atoms, ions, or molecules contained in one mole of a substance.
When dealing with gases, we use the Ideal Gas Law, represented by the formula \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature (in Kelvin).

• Pressure, volume, and temperature are needed alongside the universal gas constant to find 'n', the moles of gas.
• Conversions might be necessary to make sure units are consistent with the gas constant used.

Temperature Conversion

In order to use the Ideal Gas Law, one must convert temperature readings into Kelvin. This conversion is essential because the Kelvin scale starts at absolute zero, which is a vital consideration for scientific calculations.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature value.
For example, in the provided exercise, the temperature was given as \(22^{\circ}C\). Thus, the conversion to Kelvin is calculated as:

• \(22^{\circ}C + 273.15 = 295.15 \, K\)

This step ensures that all temperature values used in the equation are on the same scale, ensuring accurate calculations.

Pressure in atm

Pressure in the context of the Ideal Gas Law is usually measured in atmospheric pressure, or atm. The exercise provided already lists the oxygen pressure in atm, which conveniently aligns with the Ideal Gas Law's requirements.
Atmospheric pressure (atm) is a unit of measure that describes the force exerted by the atmosphere at a specific point. It's important in these calculations because it influences how gases behave in a confined space.
The standard unit for expressing pressure in gas calculations is atm, not only because it's intuitive but because it's directly usable in the Ideal Gas Law without needing conversion. This allows for straightforward input of data into the gas law formula.

Gas Volume Conversion

Volume is a measure of the space occupied by a gas, and it's expressed in liters (L) within the Ideal Gas Law. Converting any initial measurements into liters is crucial for correct calculations.
For example, gas volume may often initially be provided in milliliters (mL). To convert milliliters to liters, divide by 1000, since there are 1000 milliliters in a liter.
The problem statement gives a gas volume of \(75.0 \, mL\), so the conversion to liters is:

• \(75.0 \times \frac{1 \text{ L}}{1000 \text{ mL}} = 0.075 \text{ L}\)

Using liters as the unit of volume ensures consistency throughout the calculations and allows accurate application of the Ideal Gas Law.