- Identify the ideal gas law formula to solve for moles: \(n = \frac{PV}{RT}\).
- Convert the temperature to Kelvin: \(17.0^\circ\text{C} + 273.15 = 290.15 \text{ K}\).
- Substitute the values into the formula to find the number of moles: \(\frac{750 \times 1.35}{62.396 \times 290.15} \approx 0.0559 \text{ mol}\).
- The final number of moles is \(\boxed{0.0559 \text{ mol}}\).
Explanation
- Identifying the Correct Formula
To determine the number of moles of air, we use the Ideal Gas Law equation:
\[PV = nRT\]
where:
- \(P\) is the pressure of the gas.
- \(V\) is the volume of the gas.
- \(n\) is the number of moles of gas.
- \(R\) is the ideal gas constant.
- \(T\) is the absolute temperature of the gas in Kelvin.
We are given the following values:
- Pressure \(P = 750 \text{ torr}\)
- Volume \(V = 1.35 \text{ L}\)
- Temperature \(T = 17.0^\circ\text{C}\)
- Ideal gas constant \(R = 62.396 \text{ L} \cdot \text{torr} / (\text{mol} \cdot \text{K})\)
Our objective is to find the number of moles \(n\). We rearrange the ideal gas law to isolate \(n\):
\[n = \frac{PV}{RT}\]
Comparing this to the given choices, the correct equation is \(n = \frac{PV}{RT}\).
- Converting Units
Before performing the calculation, the temperature must be converted from Celsius to Kelvin, as the gas constant \(R\) is defined in terms of Kelvin. The conversion formula is:
\[T(K) = T(^\circ\text{C}) + 273.15\]
Substituting the given temperature:
\[T(K) = 17.0 + 273.15 = 290.15 \text{ K}\]
- Calculating the Number of Moles
Now we substitute the known values into the equation \(n = \frac{PV}{RT}\):
\[n = \frac{750 \text{ torr} \times 1.35 \text{ L}}{62.396 \text{ L} \cdot \text{torr} / (\text{mol} \cdot \text{K}) \times 290.15 \text{ K}}\]
First, calculate the numerator:
\[750 \times 1.35 = 1012.5\]
Next, calculate the denominator:
\[62.396 \times 290.15 \approx 18104.2854\]
Finally, solve for \(n\):
\[n = \frac{1012.5}{18104.2854} \approx 0.055926 \text{ mol}\]
Rounding to three significant figures, we get:
\[n \approx 0.0559 \text{ mol}\]
Examples
In meteorology and atmospheric science, the Ideal Gas Law is essential for calculating air density and understanding how pressure and temperature affect weather patterns. By determining the number of moles of air in a specific volume, researchers can calculate how much air needs to be pumped into a high-altitude weather balloon or understand the buoyancy of air masses. This fundamental relationship is used every day in environmental monitoring and aviation safety.