To determine the temperature of the air inside the pressure cooker required to just lift the whistle, we need to analyze the forces acting on the whistle and apply the ideal gas law. The whistle will lift when the pressure inside the cooker exceeds the atmospheric pressure by an amount sufficient to overcome the weight of the whistle. Let's break this down step by step.
Understanding the Forces at Play
The whistle has a weight that creates a downward force, which can be calculated using the formula:
- Weight (W) = mass (m) × gravitational acceleration (g)
Given that the mass of the whistle is 100 grams (0.1 kg) and taking g as approximately 9.81 m/s², we can calculate the weight:
- W = 0.1 kg × 9.81 m/s² = 0.981 N
Pressure Required to Lift the Whistle
The area of the whistle is given as 0.1 cm², which we need to convert to square meters for consistency in units:
- Area (A) = 0.1 cm² = 0.1 × 10⁻⁴ m² = 1 × 10⁻⁶ m²
The pressure required to lift the whistle can be calculated using the formula:
- Pressure (P) = Force (F) / Area (A)
Here, the force is equal to the weight of the whistle:
- P = W / A = 0.981 N / (1 × 10⁻⁶ m²) = 981,000 Pa = 9.81 × 10⁵ Pa
Calculating the Required Internal Pressure
For the whistle to just lift, the internal pressure (P_internal) must be greater than the atmospheric pressure (P_atmospheric) by the amount calculated:
- P_internal = P_atmospheric + P_required
- P_internal = 10⁵ Pa + 9.81 × 10⁵ Pa = 10.81 × 10⁵ Pa
Using the Ideal Gas Law
Now, we can use the ideal gas law to find the temperature required to achieve this pressure:
Where:
- P = pressure (Pa)
- V = volume (m³)
- n = number of moles of gas
- R = ideal gas constant (8.314 J/(mol·K))
- T = temperature (K)
First, we need to find the number of moles of air in the cooker at the initial conditions:
- V = 10 liters = 0.01 m³
- P = 10⁵ Pa
- Using the ideal gas law rearranged: n = PV / RT
Substituting the known values:
- n = (10⁵ Pa × 0.01 m³) / (8.314 J/(mol·K) × 300 K)
- n ≈ 0.4 moles
Finding the New Temperature
Now we can find the new temperature required to achieve the internal pressure of 10.81 × 10⁵ Pa:
- Using the ideal gas law again: T = PV / nR
Substituting the new pressure:
- T = (10.81 × 10⁵ Pa × 0.01 m³) / (0.4 moles × 8.314 J/(mol·K))
- T ≈ 325.5 K
Final Thoughts
Thus, the temperature of the air inside the pressure cooker should be approximately 325.5 K for the whistle to just lift up. This calculation illustrates how pressure, volume, and temperature are interrelated in a gas system, and how we can manipulate these variables to achieve desired outcomes in practical applications like cooking.
