Answered

An ideal gas is held in a constant volume container fitted with a valve through which gas
can escape. Starting at 250 K, the temperature of the gas is raised by 1000 K while its
pressure in the container is allowed to double.
Calculate the fraction of the original number of gas molecules that escape through the
valve.

Verified 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%.

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