The molar entropy at a specific temperature can be calculated using the equation:
ΔS = ∫ C_p / T dT
Where:
ΔS is the change in molar entropy
C_p is the molar constant-pressure heat capacity
T is the temperature
In this case, the molar constant-pressure heat capacity (C_p) is given as 0.43 J K^−1 mol^−1, and the temperature (T) is 4.2 K.
Let's calculate the molar entropy at that temperature using the equation:
ΔS = ∫ C_p / T dT
ΔS = ∫ 0.43 J K^−1 mol^−1 / T dT
Integrating this equation gives:
ΔS = 0.43 ln(T) + C
To determine the constant C, we need an initial condition. Assuming the molar entropy at absolute zero (T = 0 K) is zero, we can substitute this condition into the equation:
0 = 0.43 ln(0) + C
As ln(0) is undefined, we need to choose a very small temperature close to absolute zero, such as 0.001 K:
0 = 0.43 ln(0.001) + C
Solving this equation, we find:
C ≈ -0.43 ln(0.001)
Now we can substitute this value into the original equation to calculate the molar entropy at 4.2 K:
ΔS = 0.43 ln(T) - 0.43 ln(0.001)
ΔS = 0.43 ln(4.2) - 0.43 ln(0.001)
Using a calculator, we get:
ΔS ≈ 2.071 J K^−1 mol^−1
Therefore, the molar entropy of the solid at 4.2 K is approximately 2.071 J K^−1 mol^−1.