A sample of methane gas of mass 35 g at 250 K and 12.5 atm expands isothermally until its

pressure is 1.5 atm. Determine the change in entropy of the gas.

To determine the change in entropy of the gas, we need to calculate the final and initial entropy values and then find the difference between them.

The entropy change of an ideal gas undergoing an isothermal expansion can be calculated using the formula:

ΔS = nR ln(V₂/V₁)

Where:
ΔS = Change in entropy
n = Number of moles of gas
R = Gas constant
V₂ = Final volume of the gas
V₁ = Initial volume of the gas

To calculate the initial volume, we need to use the ideal gas law equation:

PV = nRT

Where:
P = Pressure
V = Volume
n = Number of moles of gas
R = Gas constant
T = Temperature

Rearranging the equation, we can find the initial volume:

V₁ = (nRT₁) / P₁

Similarly, for the final volume:

V₂ = (nRT₂) / P₂

Now, let's calculate the number of moles (n) using the given mass and molar mass of methane (CH₄):

n = mass / molar mass

Molar mass of CH₄ = 12.01 g/mol (for carbon) + 4(1.01 g/mol) = 16.05 g/mol

Substituting the values, we get:

n = 35 g / 16.05 g/mol ≈ 2.18 mol

Next, we need to calculate the initial and final volumes using the given information. Assuming the temperature remains constant, we can use the ideal gas law equation and substitute the values:

V₁ = (2.18 mol * 0.0821 L·atm/(K·mol) * 250 K) / 12.5 atm
V₁ ≈ 10.961 L

V₂ = (2.18 mol * 0.0821 L·atm/(K·mol) * 250 K) / 1.5 atm
V₂ ≈ 36.537 L

Now that we have the initial and final volumes, we can calculate the change in entropy:

ΔS = (2.18 mol * 0.0821 L·atm/(K·mol)) * ln(36.537 L/10.961 L)
ΔS ≈ (2.18 mol * 0.0821 L·atm/(K·mol)) * ln(3.33)
ΔS ≈ 0.455 J/K

Therefore, the change in entropy of the gas is approximately 0.455 J/K.