The cylinder in the drawing contains 3.81 mol of an ideal gas. By moving the piston, the volume of the gas is reduced to one-fourth its initial value, while the temperature is held constant. How many moles Δn of the gas must be allowed to escape through the valve, so that the pressure of the gas does not change?

its 3/4 of the original mol

(3/4) * 3.81= 2.8575

To solve this problem, we need to apply the ideal gas law, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.

Given:
Initial number of moles of gas, n1 = 3.81 mol
Initial volume of the gas, V1 = V
Final volume of the gas, V2 = V1/4 = V/4
Temperature, T1 = T2 (constant)

To find the final number of moles of gas, n2, we need to compare the initial and final states of the gas.

Applying the ideal gas law to the initial state:
P1 * V1 = n1 * R * T1

Applying the ideal gas law to the final state:
P2 * V2 = n2 * R * T2

Since the pressure does not change, P1 = P2 = P.
Also, T1 = T2, so we can simplify the equations:

P * V1 = n1 * R * T1
P * V2 = n2 * R * T2

Since T1 = T2, the temperature terms cancel out:

P * V1 = n1 * R
P * V2 = n2 * R

Suppose the number of moles of gas that must escape is Δn.

The initial number of moles can be expressed as:
Initial number of moles = n1 = n2 + Δn

Substituting n1 into the equation, we get:
P * V1 = (n2 + Δn) * R

We also know that V1 = 4 * V2, so we can substitute V2 in terms of V1 into the equation:
P * 4 * V2 = (n2 + Δn) * R

Divide both sides of the equation by R:
4 * P * V2 / R = n2 + Δn

We now need to solve for Δn:
Δn = 4 * P * V2 / R - n2

Therefore, to find the number of moles Δn of the gas that must escape through the valve, we need to know the value of the pressure P and the initial number of moles n2.