To tackle this problem, we can utilize the Ideal Gas Law, which is expressed as PV = nRT. Here, P represents pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. We will break down the solution into two parts: first, calculating the number of moles of oxygen gas, and then determining the final temperature after the gas expands.
Calculating the Number of Moles of Oxygen
We start with the initial conditions of the oxygen gas:
- Volume (V1) = 1130 cm³ = 1130 x 10-6 m³ (since 1 cm³ = 10-6 m³)
- Pressure (P1) = 101 kPa = 101,000 Pa (since 1 kPa = 1,000 Pa)
- Temperature (T1) = 42.0ºC = 42.0 + 273.15 = 315.15 K
Now, we can rearrange the Ideal Gas Law to solve for the number of moles (n):
n = PV / RT
Substituting the values into the equation:
n = (101,000 Pa) * (1130 x 10-6 m³) / (8.314 J/(mol·K) * 315.15 K)
Calculating this gives:
n = (114.153 x 10-3) / (2618.64) ≈ 0.0436 moles
Finding the Final Temperature
Next, we need to determine the final temperature (T2) after the gas expands to the new conditions:
- Volume (V2) = 1530 cm³ = 1530 x 10-6 m³
- Pressure (P2) = 106 kPa = 106,000 Pa
Using the Ideal Gas Law again, we can rearrange it to find T2:
T2 = PV / (nR)
Substituting the final conditions into the equation:
T2 = (106,000 Pa) * (1530 x 10-6 m³) / (0.0436 moles * 8.314 J/(mol·K))
Calculating this gives:
T2 = (162.18 x 10-3) / (0.3625) ≈ 447.5 K
To convert this back to Celsius:
T2 = 447.5 - 273.15 ≈ 174.35ºC
Summary of Results
In summary, we found:
- (a) The number of moles of oxygen in the system is approximately 0.0436 moles.
- (b) The final temperature after expansion is approximately 174.35ºC.
This approach illustrates how the Ideal Gas Law can be applied to different states of a gas, allowing us to derive important thermodynamic properties. If you have any further questions or need clarification on any steps, feel free to ask!
