Heat Capacity Ratio of Gases
Kaveendi Chandrasiri [email protected] Office Hrs: Thursday. 11 a.m.,
Beaupre 305 1 Purpose
• To determine the heat capacity ratio for a monatomic and a diatomic gas. • To understand and mathematically model reversible & irreversible adiabatic processes for ideal gases. • To practice error propagation for complex functions.
2 Key Physical Concepts
3 Theory: Heat Capacity
4 Theory: Determination of Heat Capacity Ratio
• We will subject a gas to an adiabatic expansion and then allow the gas to return to its original temperature via an isovolumetric process, during which time it will cool.
• This expansion and warming can be modeled in two different ways.
5 Reversible Expansion (Textbook)
• Assume that pressure in carboy (P1) and exterior pressure (P2) are always close enough that entire process is always in equilibrium
• Since system is in equilibrium, each step must be reversible
6 Irreversible Expansion (Lab Manual)
• Assume that pressure in carboy (P1) and exterior pressure (P2) are not close enough; there is sudden deviation in pressure; the system is not in equilibrium
• Since system is not in equilibrium, the process becomes irreversible.
7 Part I: Experimental Procedure Set Up of Apparatus
Use Use line A to input Ar, line B to input N2, which reaches the bottom of which does not reach the the carboy. bottom of the carboy.
Line B will be the output line. Line A will be the output line.
Nitrogen
manometer manometer C Argon C B B
A A 8 Part I: Experimental Procedure
1) Set up apparatus, insert rubber stopper. 2) Open tubes A & B, attach C to manometer. – Attach one gas to the appropriate input line. – Loosely place clamp on output line. 3) Allow gas to flow into carboy (flush system) at 15 mbar for 15 minutes. Turn off gas. 4) Close clamp. Slowly turn on the gas a very small amount while holding down stopper. 5) When manometer reaches 60 mbar turn off gas. 6) Wait until manometer reading is constant. Record manometer reading (Man1). 9 Part I: Experimental Procedure 7) Remove stopper 2-3” vertically -- Partner 1: Replace stopper tightly as quickly as possible & hold down stopper. -- Partner 2: Record lowest manometer reading (Man2). 8) Record manometer reading again when manometer reading is constant (Man3). 9) Flush system for 3 minutes. 10) Repeat steps 4 through 9 for a total of 3 measurements. 11) Switch gases, repeat experiment. -- Between runs of the same gas flush (step 3) for only 3 minutes. Between different gasses flush for 15 minutes. Why?
12) Record lab temperature & barometric pressure.10 Part II: Data Analysis • For an adiabatic expansion, three states of gas will be expressed as: – Before expansion: P1, V1, T1, n1 – Immediately after expansion: P2, V2, T2, n1 – After returning to room temperature: P3, V2, T1, n1 Pressures: • P1 = Man1 + Barometer • P2 = Man2 + Barometer • P3 = Man3 + Barometer Expressions: • Reversible: γ = [ln(P1 / P2)] / [ln(P1 / P3)] • Irreversible: γ = [(P1 / P2) – 1] / [(P1 / P3) – 1]
11 Part III: Laboratory Report
✓ Title Page: Title, name, partner(s), date of experiment. ✓ Abstract: One (1) paragraph of what, why, how, and results. ✓ Introduction: Discussion of purpose and general nature of experiment, derive expansion equations. ✓ Theory: State all assumptions, define all variables, give variations on formulas. – Knowledge: Compare and contrast reversible & irreversible adiabatic expansion expressions. – Hypothesis: Explain which you expected to closest resemble this experiment & why. ✓ Procedure & Original Data: Both signed. ✓ Results Table: All data & calculated values.
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MAT/2014 Part III: Laboratory Report
✓ Calculations: (SHOW ALL WORK) • Calculate theoretical heat capacity for each gas. • For each trial, determine γ for both expressions. • Include at least one sample calculation of each type of calculation used in numerical analysis.
✓ Error Analysis: (SHOW ALL WORK) • Calculate γ as the average of all trials for each gas and each expression. • Propagate errors in each pressure. • Propagate statistical error in γ for each expression. • Choose one trial to calculate error in γ for each expression; identify which trial is used. • Assume error of ±2 in the last recorded figure of manometer and barometer. 13
MAT/2014 Part III: Lab Report Error Analysis
Propagate error in each of the following: • ξ (P1 )2 = ξ( Man1 ) 2 + ξ( Barometer) 2 2 2 2 Use General • ξ (P2 ) = ξ (Man2 ) + ξ (Barometer) Rule • ξ (P3 ) 2 = ξ (Man3 ) 2 + ξ (Barometer) 2
•Reversible: ξ γ = [ln(P1 / P2)] / [ln(P1 / P3)] Need to •Irreversible: ξ γ = [(P1 / P2) – 1] / [(P1 / P3) – 1] derive
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MAT/2014 Part III: Laboratory Report • Summary of Data: For each gas, report both expressions with associated error [γ±ε(γ)] and correct significant figures, and indicate which is the best expression and why. • Conclusions: – Discuss significance of results. Do they match your hypothesis? – Compare γ±ε(γ) for both reversible and irreversible expansions. Do the two values lie within their respective statistical errors? – Compare γ±ε(γ) with the expected values for a monatomic and diatomic gas. Do the theoretical results lie outside the experimental errors? If so, explain plausible reasons for the lack of agreement. – Would you be able to tell the monatomic and diatomic gases apart based solely on your data? Explain. 15 Important Reminders
• Use line A for Argon input. • Use line B for N2 input. • Make sure one partner is holding stopper tightly in carboy from time = 0s. • When calculating pressures, add manometer readings with barometric pressure of room. • TA will only sign data AFTER you have
closed both gases. 16