The molar constant-pressure heat capacity of a certain solid at 4.2 K is 0.43 J K−1 mol−1. What is its molar entropy at that temperature?

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.

To calculate the molar entropy of a substance at a specific temperature, we need to use the equation:

ΔS = ∫ (Cp / T) dT

Where:
ΔS = change in molar entropy
Cp = molar constant-pressure heat capacity
T = temperature

First, we need to find the integral of (Cp / T) with respect to T. The integral of (Cp / T) is given by:

ΔS = Cp * ln(T2 / T1)

Where:
T2 = final temperature
T1 = initial temperature

In this case, the initial temperature is 4.2 K, and we need to find the molar entropy at that temperature. Let's assume the final temperature is T2. Therefore, the equation becomes:

ΔS = 0.43 J K−1 mol−1 * ln(T2 / 4.2 K)

To find the molar entropy at 4.2 K, we need to substitute 4.2 K as the final temperature (T2) into the equation:

ΔS = 0.43 J K−1 mol−1 * ln(4.2 K / 4.2 K)

Simplifying further:

ΔS = 0.43 J K−1 mol−1 * ln(1)

The natural logarithm of 1 is 0, so we find:

ΔS = 0.43 J K−1 mol−1 * 0

Therefore, the molar entropy at 4.2 K for the given solid is 0 J K−1 mol−1.