- Use the Ideal Gas Law: \(PV = nRT\).
- Convert temperature to Kelvin: \(T = 20.0 + 273.15 = 293.15 \, \text{K}\).
- Rearrange the formula for volume: \(V = \frac{nRT}{P}\).
- Calculate the volume using the given values: \(V = \frac{2.0 \times 62.396 \times 293.15}{720} \approx 50.8 \, \text{L}\), which rounds to \(\boxed{51 \, \text{L}}\).
Explanation
- Analyzing the Problem and Preparing Data
To find the volume of nitrogen gas, we use the Ideal Gas Law equation, which describes the relationship between the pressure (\(P\)), volume (\(V\)), amount of substance in moles (\(n\)), the universal gas constant (\(R\)), and temperature (\(T\)). The equation is given as:
\[P V = n R T\]
We are given the following values:
- Number of moles \(n = 2.0 \, \text{mol}\)
- Pressure \(P = 720 \, \text{torr}\)
- Temperature in Celsius \(T_{Celsius} = 20.0^\circ\text{C}\)
- Gas constant \(R = 62.396 \, \text{L}\cdot\text{torr}/\text{mol}\cdot\text{K}\)
Before we can use these values, we need to convert the temperature from Celsius to Kelvin, as the Ideal Gas Law requires absolute temperature. We use the conversion formula \(T_K = T_{Celsius} + 273.15\):
\[T = 20.0 + 273.15 = 293.15 \, \text{K}\]
- Calculating the Volume of Nitrogen Gas
Now that we have all the required parameters in the correct units, we can rearrange the Ideal Gas Law to solve for the volume \(V\):
\[V = \frac{n R T}{P}\]
Substituting the known values into the equation:
\[V = \frac{2.0 \, \text{mol} \times 62.396 \, \text{L}\cdot\text{torr}/\text{mol}\cdot\text{K} \times 293.15 \, \text{K}}{720 \, \text{torr}}\]
First, calculate the product of the numerator:
\[2.0 \times 62.396 \times 293.15 = 36582.8558 \, \text{L}\cdot\text{torr}\]
Now, divide by the pressure \(P = 720 \, \text{torr}\) to find the volume:
\[V = \frac{36582.8558}{720} \approx 50.809 \, \text{L}\]
Comparing this result to the given options, we find that the volume is approximately \(51 \, \text{L}\).
- Final Conclusion
Based on our calculation, the volume of 2.0 moles of nitrogen gas at 720 torr and 20.0°C is approximately 50.8 L. Rounding this value to the nearest provided option, we select 51 L. Thus, the final answer is \(\boxed{51 \, \text{L}}\).
Examples
The Ideal Gas Law is a fundamental principle used in chemistry and physics to understand how gases behave under different conditions. In real life, this is used in fields such as engineering, where it might be necessary to calculate the volume of gas needed to fill a container at a specific pressure and temperature, or in medical fields for calculating oxygen flow in ventilators. By using this law, scientists and engineers can predict and control the behavior of gases in various controlled environments, ensuring safety and precision.