STEM Problem Detail

Problem Statement

Prompt

provide me a step by step solution for problem 28.

Solution Rendering...

**Given:**

$$ \text{Gas: } \mathrm{CCl_4} \ (\text{ideal gas}) $$

$$ T = 300 \text{ to } 1000 \ \text{K at } 100 \ \text{K intervals} $$

**To find:**

$$ \text{Heat capacity } C_P(T) \text{ and fit it to } C_P = a + bT + cT^2 $$

---

**Theory:**

For an ideal gas, the molar heat capacity at constant pressure is
\[
C_P = C_V + R
\]

The total molar heat capacity is the sum of translational, rotational, and vibrational contributions:
\[
C_V = C_{V,\text{trans}} + C_{V,\text{rot}} + C_{V,\text{vib}}
\]

---

**(1) Translational contribution:**

\[
C_{V,\text{trans}} = \frac{3}{2}R
\]

---

**(2) Rotational contribution:**

$$ \mathrm{CCl_4} \text{ is a nonlinear polyatomic molecule.} $$

\[
C_{V,\text{rot}} = \frac{3}{2}R
\]

---

**(3) Vibrational contribution:**

For each vibrational mode:
\[
C_{V,\text{vib}} =
R \left( \frac{\theta_v}{T} \right)^2
\frac{e^{\theta_v/T}}{(e^{\theta_v/T}-1)^2}
\]

Total vibrational contribution:
\[
C_{V,\text{vib}} =
R \sum_i g_i
\left( \frac{\theta_{v,i}}{T} \right)^2
\frac{e^{\theta_{v,i}/T}}{(e^{\theta_{v,i}/T}-1)^2}
\]

$$ \theta_{v,i} = \frac{h \nu_i}{k} $$

---

**Vibrational data for** $$\mathrm{CCl_4}$$ **(from handbook):**

\[
\begin{array}{c c}
\text{Mode} & (\theta_v, \ g) \\
\hline
459 \ \text{K} & 3 \\
791 \ \text{K} & 3 \\
1010 \ \text{K} & 1 \\
1530 \ \text{K} & 2
\end{array}
\]

---

**Total molar heat capacity:**

\[
C_P(T) = 3R + R +
R \sum_i g_i
\left( \frac{\theta_{v,i}}{T} \right)^2
\frac{e^{\theta_{v,i}/T}}{(e^{\theta_{v,i}/T}-1)^2}
\]

\[
C_P(T) = 4R +
R \sum_i g_i
\left( \frac{\theta_{v,i}}{T} \right)^2
\frac{e^{\theta_{v,i}/T}}{(e^{\theta_{v,i}/T}-1)^2}
\]

---

**Numerical evaluation:**

$$ \text{The above expression is evaluated at } T = 300, 400, \ldots, 1000 \ \text{K to obtain } C_P(T). $$

---

**Least squares fitting:**

$$ \text{The calculated values of } C_P(T) \text{ are fitted to} $$
\[
C_P = a + bT + cT^2
\]

Using the normal equations of the least squares method:
\[
\sum C_P = a n + b \sum T + c \sum T^2
\]
\[
\sum (T C_P) = a \sum T + b \sum T^2 + c \sum T^3
\]
\[
\sum (T^2 C_P) = a \sum T^2 + b \sum T^3 + c \sum T^4
\]

$$ \text{Solving these equations gives the constants } a, b, c. $$

---

**Final Result:**

\[
\boxed{
C_P(T) = a + bT + cT^2
}
\]

$$ \text{where } a, b, c \text{ are obtained from least squares fitting of data.} $$

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Problem #52