Final answer:
Calculating the entropy change of helium gas using the entropy formula, the value of x in the expression (2.303 x x) cal should be 4 cal/mol·K, which is not listed in the options. Thus, we should refrain from providing an answer.
Step-by-step explanation:
The change in entropy (ΔS) for a mole of helium gas changing states can be found using the formula:
ΔS = nR ln(V2/V1) + nR ln(T2/T1)
Since the volume change is not given directly but the pressure change is, and assuming ideal gas behavior, we can use the inverse relationship between pressure and volume (P1V1 = P2V2) to replace the volume ratio with the pressure ratio (V2/V1 = P1/P2) as V2/V1 is inversely proportional to P1/P2 for a constant amount of gas at constant temperature.
Thus,
ΔS = nR ln(P1/P2) + nR ln(T2/T1)
Plugging in the given values:
ΔS = (1 mol)(2 cal/mol·K) ln(10 atm / 1 atm) + (1 mol)(2 cal/mol·K) ln(1000 K / 100 K)
This simplifies to:
ΔS = 2 cal/mol·K ln(10) + 2 cal/mol·K ln(10)
ΔS = 2 cal/mol·K (2.303) + 2 cal/mol·K (2.303)
ΔS = 4.606 cal/mol·K + 4.606 cal/mol·K
ΔS = 9.212 cal/mol·K
The change in entropy is given as (2.303 × x) cal, to find x we have:
9.212 cal/mol·K = 2.303 × x
x = 9.212 / 2.303
x = 4 cal/mol·K, which means the answer is not one of the options provided. Since the values from the question might be rounded, we need to identify the closest value to our result, which is 4 cal/mol·K.
Therefore, we should not answer when we are not confident in the correctness of any part of our answer.