Answer :
The fraction of gas molecules that escaped is 80%. This was calculated using the Ideal Gas Law and the given temperature and pressure changes. The fraction of remaining molecules was found to be 1/5, leading to 4/5 escaping.
To solve this problem, you need to apply the Ideal Gas Law, which is given by the formula PV = NkT, where P is pressure, V is volume, N is the number of molecules, k is Boltzmann's constant (1.38×10⁻²³J/K), and T is temperature.
Initially, the gas is at P₁, V, T₁, N₁. After heating, it is at P₂, V, T₂, N₂
Given:
The Ideal Gas Law before and after the change:
Initial state: P₁V = N₁kT₁ which simplifies to P₁= N₁kT₁/V.
Final state: P₂V = N₂kT₂ which simplifies to P₂ = N₂kT₂/V.
From the given, P₁V = P = N₁kT₁ , and P₂ = 2P ₁= N₂kT₂./V
By Ratio: (N₁ * T₁ ) / N₂ * T₂ = 1 / 2.
Substitute T₁ and T₂:
(N₁ * 250) = N₂ * 1250 / 2.
N₂ = (N₁* 250) / (1250 / 2)
N₂ = (N₁ * 250) / 625
N₂ = 2N₁ / 5
The fraction of gas molecules remaining is N₂ / N
= (2N / 5) / N
= 2/5.
Thus, the fraction of molecules that escaped is 1 - (2/5) = 3/5 or 0.6 (60%).
Approximately 60% of the original gas molecules escape through the valve when the temperature is raised by 1000 K and the pressure is allowed to double, assuming constant volume.
To analyze the fraction of original gas molecules that escape from the container, we'll use the ideal gas law:
PV = NkT
For the initial state,
where N is the number of gas molecules,
P is the pressure,
V is the volume, and
T is the temperature:
Initial state: PV = NkT
Given the following:
- Initial Temperature (T1) = 250 K
- Final Temperature (T2) = 250 K + 1000 K = 1250 K
- Initial Pressure (P1) = P
- Final Pressure (P2) = 2P
Since the volume is constant, we can write the initial and final states of the gas:
- Initial state: P * V = N * k * 250
- Final state: 2P * V = (N - n ) * k * 1250 where n is the number of molecules that escaped.
From the initial state equation:
PV = NkT
P * V = N * k * 250
From the final state equation:
2P * V = (N - n ) * k * 1250
Dividing the final state equation by the initial state equation to eliminate the constants:
2 * (PV) / (PV) = (N - n) * 1250 / (N * 250) where n is the number of molecules that escaped.
2 = (N - n) * 5 / N
2N = 5(N - n)
2N = 5N - 5n
3N = 5n
n / N = 3 / 5 = 0.6
Therefore, the fraction of the original number of gas molecules that escape through the valve is 0.6 or 60%.
