- Identify the given variables: \(P = 2.00 \text{ atm}\), \(V = 3.00 \text{ L}\), \(T = 25.0 ^\circ\text{C}\), and \(R = 0.0821 \frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}}\).
- Convert temperature to Kelvin: \(T = 25.0 + 273.15 = 298.15 \text{ K}\).
- Use the ideal gas law formula rearranged for moles: \(n = \frac{PV}{RT}\).
- Compute the final result: \(n = \frac{2.00 \times 3.00}{0.0821 \times 298.15} \approx \boxed{0.245 \text{ moles}}\).
Explanation
- Data Analysis and Ideal Gas Law Formula
The problem requires us to determine the number of moles \(n\) of oxygen gas contained in a given volume at a specific pressure and temperature. We are given the following information:
- Pressure (\(P\)) = \(2.00 \text{ atm}\)
- Volume (\(V\)) = \(3.00 \text{ L}\)
- Temperature (\(T_C\)) = \(25.0 ^\circ\text{C}\)
- Universal gas constant (\(R\)) = \(0.0821 \frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}}\)
The relationship between these variables is described by the ideal gas law, which is given by the equation:
\[PV = nRT\]
To find the number of moles \(n\), we need to rearrange this equation to:
\[n = \frac{PV}{RT}\]
Before performing the calculation, we must ensure all units are compatible. Specifically, the temperature must be in Kelvin.
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- Temperature Conversion to Kelvin
The temperature provided is in degrees Celsius. To convert this to Kelvin (\(T_K\)), we use the standard conversion formula:
\[T_K = T_C + 273.15\]
Substituting the given temperature:
\[T_K = 25.0 + 273.15 = 298.15 \text{ K}\]
This temperature is now ready to be used in the ideal gas law calculation.
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- Calculating the Number of Moles
With all variables converted to the appropriate units, we can calculate the number of moles \(n\) using the rearranged ideal gas law:
\[n = \frac{P \times V}{R \times T_K}\]
Substituting the known values:
\[n = \frac{2.00 \times 3.00}{0.0821 \times 298.15}\]
First, calculate the numerator and the denominator:
- Numerator: \(2.00 \times 3.00 = 6.00\)
- Denominator: \(0.0821 \times 298.15 \approx 24.478\)
Now, perform the division:
\[n \approx \frac{6.00}{24.478} \approx 0.2451 \text{ moles}\]
Thus, the amount of oxygen gas required is approximately \(0.245 \text{ moles}\).
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Examples
The ideal gas law is used extensively in fields like chemistry and engineering to understand the behavior of gases under various conditions. For instance, in an industrial setting, engineers use this law to determine the amount of gas required for a storage tank or to calculate the pressure that a gas will exert when it is heated in a closed container. By knowing the relationship between volume, pressure, and temperature, we can safely and efficiently handle gases in applications ranging from industrial production to medical equipment such as oxygen tanks.