mw_of_liquid-problem-statement.org

Intro to Error Analysis

Elementary Propagation

\begin{align} δ A &= \sqrt{\left( \frac{∂ A}{∂ a}\right) ^2 _bM δ a^2 + \left(\frac{∂ A}{∂ b} \right)^2_aM δ b^2 + \left(\frac{∂ A}{∂ M}\right)^2_ab δ M^2 } \notag

&= \sqrt{(b^2 M^2 δ a^2 + a^2 M^2 δ b^2 + a^2 b^2 δ M^2)} \notag %

\end{align} Derive uncertainty relationships for the following.

1. Root-mean-square speed of a molecule; $R$ constant. \[ v_\text{rms} = \sqrt{\frac{3 RT}{M}} \]
2. An equation of state for gases; $R$ constant \[ P = \frac{RT}{V-b} - \frac{a}{V^2} \]
3. Compressibility of a gas; $R$ constant \[ Z = \frac{PV}{RT} \]
4. Rotational partition function; $k, h, σ$ constant

   \[ q_r = \frac{2IkT}{σ h^2} \]

5. Translational partition function; $k, h$ constant \[ q_tr = \left( \frac{2 π mkT}{h^2}\right) ^3/2\]

An example: The molecular mass of a volatile liquid

Estimate the propagated uncertainty for an experiment from an old lab experiment (Determination of the Molecular Weight of a Volatile Liquid) by reconstructing on paper a plausible experiment. Assume that you use an analytical balance, graduated cylinder, barometer, and ordinary thermometer. Report the uncertainty and comment on the largest source of error, any discrepancies, etc.

Follow best practices for reproducible work. For example, if using a spreadsheet, use informative labels, arrange columns to indicate the flow of the calculation, use absolute referencing for constants, and the like. If using python, choose informative variable names, use explanatory text, think about the flow of the calculation.

With either computational tool, imagine an outsider trying to understand the work…s/he is your audience. If you’re not sure about these practices, contact me so we can go over them together.

The student data is available in the file mw_of_liquid_student_data.pdf.