STEM Problem Detail

Problem Statement

Prompt

provide me a step by step solution for problem 40.

Solution Rendering...

**Given:**

$$ \text{Gas: } \mathrm{Cl_2} $$

$$ T = 298 \ \text{K} $$

$$ P = 1.0133 \times 10^5 \ \text{Pa} $$

$$ r = 1.988 \times 10^{-10} \ \text{m} $$

$$ \tilde{\nu} = 0.5649 \times 10^5 \ \text{m}^{-1} $$

$$ q_{\text{elec}} = 1 $$

**To find:**

\[
\frac{G^\circ - H_0^\circ}{T}
\]

---

**Theory:**

The Gibbs free energy function for an ideal gas is given by
\[
\frac{G^\circ - H_0^\circ}{T}
= -R \ln q + RT \left( \frac{\partial \ln q}{\partial T} \right)_V
\]

The total molecular partition function is
\[
q = q_{\text{trans}} q_{\text{rot}} q_{\text{vib}} q_{\text{elec}}
\]

---

**(1) Translational contribution:**

\[
q_{\text{trans}} =
\left( \frac{2 \pi m k T}{h^2} \right)^{3/2}
\frac{kT}{P}
\]

$$ \text{Molar mass of } \mathrm{Cl_2} = 70.9 \ \text{g mol}^{-1} $$

\[
m = \frac{70.9 \times 10^{-3}}{6.022 \times 10^{23}}
= 1.18 \times 10^{-25} \ \text{kg}
\]

---

**(2) Rotational contribution:**

For a diatomic molecule:
\[
q_{\text{rot}} = \frac{T}{\sigma \theta_r}
\]

$$ \sigma = 2 $$

Moment of inertia:
\[
I = \mu r^2
\]

\[
\mu = \frac{m}{2}
\]

\[
\theta_r = \frac{h^2}{8 \pi^2 I k}
\]

---

**(3) Vibrational contribution:**

\[
\theta_v = \frac{h c \tilde{\nu}}{k}
\]

\[
q_{\text{vib}} =
\frac{1}{1 - e^{-\theta_v/T}}
\]

---

**(4) Electronic contribution:**

\[
q_{\text{elec}} = 1
\]

---

**Total partition function:**

\[
q = q_{\text{trans}} q_{\text{rot}} q_{\text{vib}}
\]

---

**Evaluation of free energy function:**

The Gibbs free energy function becomes
\[
\frac{G^\circ - H_0^\circ}{T}
= -R \ln q
+ RT \left( \frac{\partial \ln q}{\partial T} \right)
\]

Separating contributions:
\[
\frac{G^\circ - H_0^\circ}{T}
=
\left( \frac{G^\circ - H_0^\circ}{T} \right)_{\text{trans}}
+
\left( \frac{G^\circ - H_0^\circ}{T} \right)_{\text{rot}}
+
\left( \frac{G^\circ - H_0^\circ}{T} \right)_{\text{vib}}
\]

---

**Final Answer:**

\[
\boxed{
\left( \frac{G^\circ - H_0^\circ}{T} \right)_{\mathrm{Cl_2},\,298\,\text{K}}
\ \text{is obtained by summing translational, rotational, and vibrational contributions}
}
\]

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Problem #50