STEM Problem Detail

Problem Statement

Prompt

provide me a step by step solution for problem 9.

Solution Rendering...

**Given:**

Determine the translational component of the sum over states for methane at
$$T = 1000\ \text{K}$$ and $$P = 1.0133\times10^{5}\ \text{Pa}$$.

---

**Theory:**

For an ideal gas, the total molecular partition function can be written as a
product of individual contributions. The translational component of the
molecular sum over states per molecule is given by

$$q_{\text{trans}} = \left(\dfrac{2\pi m k T}{h^{2}}\right)^{3/2} V$$

For one mole of gas, the volume $$V$$ is obtained from the ideal gas equation,

$$V = \dfrac{RT}{P}$$

---

**Step 1: Molecular mass of methane**

The molar mass of methane is

$$M = 16.04\ \text{g mol}^{-1} = 16.04\times10^{-3}\ \text{kg mol}^{-1}$$

The mass of one molecule is

$$m = \dfrac{M}{N_A}$$

---

**Step 2: Volume at given conditions**

Using the ideal gas equation,

$$V = \dfrac{RT}{P}$$

Substituting values,

$$V = \dfrac{(8.314)(1000)}{1.0133\times10^{5}}$$

---

**Step 3: Translational sum over states**

Substituting $$m$$ and $$V$$ into the translational expression,

$$q_{\text{trans}}
= \left(\dfrac{2\pi m k T}{h^{2}}\right)^{3/2}
\left(\dfrac{RT}{P}\right)$$

---

**Final expression at $$T = 1000\ \text{K}$$:**

$$\boxed{
q_{\text{trans}}
= \left(\dfrac{2\pi m k (1000)}{h^{2}}\right)^{3/2}
\left(\dfrac{R(1000)}{1.0133\times10^{5}}\right)
}$$

---

**Constants used:**

$$k$$ is the Boltzmann constant,
$$h$$ is Planck’s constant,
$$R$$ is the gas constant,
$$N_A$$ is Avogadro’s number.

Loading solution...

Problem #94