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Chapter 12: Problem 3

Residual molar entropy is given by the expression (a) \(S=R \ln W\) (b) \(S=N \ln W\) (c) \(S=\frac{R}{N} \ln W\) (d) \(S=n R \ln W\)

Short Answer

Expert verified
The correct expression for the residual molar entropy is \(S = nR \ln W\), so option (d) is the answer.

Step by step solution

01

Recall Boltzmann's Entropy Formula

The correct formula for entropy according to Boltzmann's entropy formula is given by \(S = k \ln W\), where \(k\) is the Boltzmann constant and \(W\) is the number of microstates.
02

Relate Boltzmann Constant to Gas Constant

The Boltzmann constant \(k\) can be written in terms of the gas constant \(R\) and Avogadro's number \(N_A\), as \(k = R/N_A\). Here, \(N_A\) is Avogadro's number which is the number of atoms per mole. The entropy equation becomes \(S = \frac{R}{N_A}\ln W\).
03

Express Entropy in Terms of Moles

In molar terms, \(N_A\) is replaced by the number of moles \(n\), leading to \(S = nR \ln W\).

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Most popular questions from this chapter

The third law of thermodynamics may be stated as follows: (a) At absolute zero the entropy of all perfectly crystalline solids tends to decrease. (b) The entropy of a perfectly crystalline solid may be taken as zero at absolute zero. (c) The entropy of every substance is zero at \(0 \mathrm{~K}\). (d) The entropy of a substance is related to its heat capacity.

From the third law, the entropy of a substance at temperature \(T\) is given as, (a) \(S_{T}=\frac{1}{3} C_{P}\left(\right.\) at \(\left.T_{1}\right)+\int_{T_{1}}^{T} \frac{C_{p}}{T} d T\) (b) \(S=C_{p}\left(\right.\) at \(\left.T_{1}\right)+\int_{\tau_{1}}^{T} C_{p} \ln d T\) (c) \(S=C_{p} \ln T+\int_{\tau_{1}}^{T} \frac{C_{p}}{T} d T\) (d) none of the above

The Boltzmann entropy equation is (a) \(S=\frac{K}{\ln W}\) (b) \(S=K^{r} \ln \frac{P_{2}}{P_{\mathrm{J}}}\) (c) \(S=K^{\prime} \ln \frac{V_{\mathrm{J}}}{V_{2}}\) (d) \(S=K^{\prime \prime} \ln W\)

The Nernst heat theorem can be mathematically stated as (a) \(\lim _{T \rightarrow 0} \frac{d(\Delta G)}{d T}=\lim _{T \rightarrow 0} \frac{d(\Delta H)}{d T}=0\) (b) \(\lim _{T \rightarrow 0} \frac{d(\Delta H)}{d T}=\lim _{T \rightarrow 0} \frac{d(\Delta S)}{d T}=0\) (c) \(\lim _{T \rightarrow 0} \frac{d(\Delta V)}{d T}=\lim _{T \rightarrow 0} \frac{d(\Delta A)}{d T}=0\) (d) \(\lim _{r \rightarrow 0} \frac{d(\Delta A)}{d T}-\lim _{r \rightarrow 0} \frac{d(\Delta H)}{d T}\)
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