- Identify the initial and final states of the gas: \(P_1 = 1.5 \text{ atm}\), \(V_1 = 3.0 \text{ L}\), \(T_1 = 293 \text{ K}\), \(P_2 = 2.5 \text{ atm}\), \(T_2 = 303 \text{ K}\).
- Convert temperatures to Kelvin: \(T_1 = 293 \text{ K}\) and \(T_2 = 303 \text{ K}\).
- Apply the combined gas law: \(V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}\).
- Calculate the new volume: \(V_2 = \frac{1.5 \times 3.0 \times 303}{2.5 \times 293} \approx \boxed{1.86 \text{ L}}\).
Explanation
- Analysis of Gas Conditions
To find the new volume of the nitrogen gas, we start by identifying the initial and final states of the gas. The initial conditions are a pressure \(P_1 = 1.5 \text{ atm}\), a volume \(V_1 = 3.0 \text{ L}\), and a temperature \(T_{1\_C} = 20^{\circ} \text{C}\). The final conditions are a pressure \(P_2 = 2.5 \text{ atm}\), a final temperature \(T_{2\_C} = 30^{\circ} \text{C}\), and an unknown final volume \(V_2\). We use the combined gas law formula \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\), noting that temperature must be in Kelvin.
- Temperature Conversion
The ideal gas law requires the absolute temperature in Kelvin. We convert the Celsius temperatures to Kelvin using the relation \(T_K = T_C + 273\).
For the initial temperature: \(T_1 = 20 + 273 = 293 \text{ K}\).
For the final temperature: \(T_2 = 30 + 273 = 303 \text{ K}\).
- Calculation of Final Volume
To find the unknown volume \(V_2\), we rearrange the combined gas law \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\) to isolate \(V_2\):
\[V_2 = \frac{P_1 \cdot V_1 \cdot T_2}{P_2 \cdot T_1}\]
Now, we substitute the known values into the equation:
\[V_2 = \frac{1.5 \text{ atm} \cdot 3.0 \text{ L} \cdot 303 \text{ K}}{2.5 \text{ atm} \cdot 293 \text{ K}}\]
Performing the multiplication in the numerator: \(1.5 \cdot 3.0 \cdot 303 = 1363.5\).
Performing the multiplication in the denominator: \(2.5 \cdot 293 = 732.5\).
Calculating the final value: \(V_2 = \frac{1363.5}{732.5} \approx 1.8614 \text{ L}\). Rounding to a reasonable number of decimal places, we get approximately \(1.86 \text{ L}\).
Examples
The combined gas law is essential for fields like meteorology and engineering. For example, weather balloons expand as they rise into the atmosphere where pressure decreases. By using the combined gas law to account for changing pressure and temperature, scientists can accurately predict how these balloons change in size, which is critical for the equipment they carry. This principle also applies to understanding how car tires might change pressure due to temperature fluctuations, ensuring vehicle safety.