Question

Given that 6.9 moles of carbon monoxide gas are present in a container of volume \(30.4 \mathrm{~L},\) what is the pressure of the gas (in atm) if the temperature is \(82^{\circ} \mathrm{C}\) ?

Short answer

The pressure of the gas is approximately 6.55 atm.

Step-by-step solution

- Convert Celsius to Kelvin

First, we need to convert the temperature from degrees Celsius to Kelvin since gas laws use absolute temperature. The formula to convert Celsius to Kelvin is: \[\text{Temperature in Kelvin} = \text{Temperature in Celsius} + 273.15\]Calculating this:\[82^{\circ} C + 273.15 = 355.15 \, \text{K}\]

- Use the Ideal Gas Law

Next, apply the Ideal Gas Law which is given by:\[PV = nRT\]where:- \(P\) is the pressure in atmospheres,- \(V\) is the volume in liters,- \(n\) is the number of moles of gas,- \(R\) is the ideal gas constant (0.0821 L·atm/mol·K),- \(T\) is the temperature in Kelvin.

- Solve for Pressure

We need to solve for pressure \(P\). Rearrange the Ideal Gas Law formula to solve for \(P\):\[P = \frac{nRT}{V}\]Now substitute the values:- \(n = 6.9\) moles,- \(R = 0.0821\) L·atm/mol·K,- \(T = 355.15\) K,- \(V = 30.4\) L.\[P = \frac{6.9 \times 0.0821 \times 355.15}{30.4}\]

- Calculate the Pressure

Calculate the expression obtained:\[P = \frac{6.9 \times 0.0821 \times 355.15}{30.4} \approx 6.55\] atm.

Key concepts

Temperature Conversion

When dealing with gases, it's important to use the correct temperature scale to ensure your calculations are accurate. In the context of the Ideal Gas Law, we use the Kelvin scale. This is because Kelvin is an absolute temperature scale, starting at absolute zero, where theoretically all molecular motion stops. This makes it a natural choice for thermodynamic calculations.
To convert temperature from Celsius to Kelvin is simple. Use the formula: \\[ \text{Temperature in Kelvin} = \text{Temperature in Celsius} + 273.15 \] \As seen in the problem, the given temperature is \(82^\circ \mathrm{C}\). By applying the conversion formula, we add 273.15 to the Celsius temperature, resulting in 355.15 K. This Kelvin temperature is what you will use in all equations involving gas laws.
Understanding this conversion is crucial because Kelvin temperatures allow direct proportionality seen in the original laws of thermodynamics, leading to simpler and more consistent results in scientific analysis.

Pressure Calculation

In order to determine the pressure of a gas within a container, you can use the Ideal Gas Law, which is expressed as \\[PV = nRT\] \Here the symbols represent their respective quantities:

• \(P\): Pressure (in atmospheres)
• \(V\): Volume (in liters)
• \(n\): Number of moles of gas
• \(R\): Ideal gas constant (0.0821 L·atm/mol·K)
• \(T\): Temperature (in Kelvin)

With the Ideal Gas Law, rearrange the formula to solve for pressure \(P\): \\[P = \frac{nRT}{V}\] \This equation makes clear that the pressure of a gas is directly proportional to the number of moles and temperature, but inversely proportional to volume. Plug in the given values: the number of moles \(n = 6.9\), the ideal gas constant \(R = 0.0821\), the temperature \(T = 355.15\) K, and the volume \(V = 30.4\) L to find that \(P \approx 6.55\ atm\).
Understanding this relationship is vital because it illustrates how changes in any of these variables impact pressure, allowing predictions about behavior under different conditions.

Moles and Volume Relationships

One of the fascinating aspects of gas behavior is how moles and volume relate to each other. The Ideal Gas Law, \[PV = nRT\], illustrates this relationship and shows how these factors combine to determine the state of a gas within a container.
Here, 'moles' refers to the number of particles or molecules in the gas. This is significant because it provides a count that is directly usable in equations. The volume in this equation is the space that the gas molecules occupy. According to Avogadro’s Law, equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules. This principle allows us to expect that if you increase the number of moles (gas amount) without changing the volume, pressure should increase proportionally.
For example, in the exercise, 6.9 moles of gas occupies a 30.4 L container. At a constant temperature, if we were to double the number of moles to 13.8, while keeping the volume constant, the pressure should theoretically double too, provided temperature and volume remain unchanged, demonstrating a direct proportion. Understanding this makes the principles of gas behavior under varying conditions more intuitive.