Heat Capacity Ratio of Gases

Home , Adiabatic process, Isochoric process

Carson Hasselbrink [email protected] Office Hours: Mon 10-11am, Beaupre 360

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 • Heat capacity is the amount of heat required to raise the temperature of an object or substance one degree 풅풒 푪 = 풅푻 • Heat Capacity Ratio is the ratio of specific heats at constant pressure and constant volume 퐶푝 훾 = 퐶푣

휕퐻 휕퐸 Where 퐶 = ( ) and 퐶 = ( ) 푝 휕푇 푝 푣 휕푇 푣 • An adiabatic process occurs when no heat is exchanged between the system and the

surroundings 3 Theory: Heat Capacity

휕퐸 • Heat Capacity (Const. Volume): 퐶 = ( ) /푛 푣,푚 휕푇 푣 3푅푇 3푅 – Monatomic: 퐸 = , so 퐶 = 푚표푛푎푡표푚푖푐 2 푣,푚 2 5푅푇 5푅 – Diatomic: 퐸 = , so 퐶 = 푑푖푎푡표푚푖푐 2 푣,푚 2 • Heat Capacity Ratio: 퐶푝,푚 = 퐶푣,푚 + 푅 퐶푝,푚 푅 훾 = = 1 + 퐶푣,푚 퐶푣,푚 푃 [ln 1 ] 푃2 – Reversible: 훾 = 푃 [ln 1 ] 푃3 푃 [ 1 −1] 푃2 – Irreversible: 훾 = 푃 [ 1 −1] 푃3 • Diatomic heat capacity > Monatomic 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 isochoric 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 Syllabus)

• 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. – Use your own words for this section. Do NOT plagiarize! ✓ Procedure & Original Data ✓ Results Table: All data & calculated values. 12

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 Use General 2 2 2 2 2 • ξ (P ) = ξ (Man ) + ξ (Barometer) Rule • ξ (P3 ) 2 = ξ (Man3 ) 2 + ξ (Barometer) 2

• Reversible: ξ γrev Need to derive • Irreversible: ξ γirrev

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