Determination of the Heat Capacity Ratio and Cv of Gases Lab Report PDF

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Determination of the Heat Capacity Ratio, ᵞ, and Cv of Gases Ashlee Perkinson February 28, 2012 Introduction This experiment aimed to experimentally determine the heat capacity ratio, γ , of three different gases: nitrogen, carbon dioxide, and helium. Once the heat capacity ratios for the three gases were determined, they were compared to theoretical and literature values. This was accomplished by measuring the changes in pressure for the adiabatic expansion of each gas. The heat capacity ratio describes the relationship between the heat capacity at constant pressure to the heat capacity at constant volume. Heat capacity is an extensive property that depends on the amount of substance in the sample [1]. Differences in heat capacities arise due to differences in the separations of their energy levels, according to quantum theory. Closely spaced energy levels correlate with a high heat capacity because little temperature is needed to accommodate incoming energy [1]. In this experiment we will be using Charles’s Law to relate temperature and pressure, as the final temperature after the gas has cooled is not known. This is accomplished in an adiabatic system, so there is no heat exchange and work can be directly correlated with internal energy to determine the heat capacity of each gas and the heat capacity ratio.

Theory The first law of thermodynamics tells us that the relationship between heat capacities at constant volume and constant pressure is defined by Equation 1, listed below. C p −C v =R Eq. 1

The heat capacity due to volume, CV may be calculated theoretically by means of statistical thermodynamics, which accounts for shape and degrees of freedom. Degrees of freedom are comprised of translational, vibrational, and rotational ranges of motion. All atoms contain translational degrees of freedom, and molecules containing more than one atom contain rotational and vibrational degrees of freedom. There are three independent coordinates necessary to describe translational motion. Rotational and vibrational motions depend on the type of molecule and shape respectively. Statistical thermodynamics allows us to express the total heat capacity under constant volume conditions as the sum of the energies of the various types of motion, seen in Equation 2. Trans

C v =C v

+C v +C v Rot

Vib

Eq. 2

The first law of thermodynamics also states that the internal energy of a substance can be described as below in Equation 3. By taking the derivations of this equation and pairing it with the definitions of the heat capacities of substances, as listed in Equation 4 below, we can derive an expression for the heat ration. dU =dq + dw Eq. 3

C p =( ᵨH / ᵨT )P and C v =( ᵨU / ᵨT )v Eq. 4 Because the process is adiabatic and there is no exchange of heat, dq=0. Therefore, Equation 5 can be used to relate work and internal energy. Work is also described as the product of negative external pressure and the change in volume of a system; internal energy is described as the product of heat capacity and change in temperature. Equation 6 describes this new relationship between internal energy and work. dU =dw Eq. 5

C v dT =− p ext dV

Eq. 6

By integrating Equation 6 and replacing external pressure with p2, which stays constant during the experiment, the relationship in Equation 7 can be obtained. Because of the nature of this experiment, we want to redefine the equations in terms of pressure. To do this, V1 and V2 can be replaced with its definition from the ideal gas law equation, seen in Equation 8. Because R was previously defined in Equation 1, it can be replaced to yield the expression in Equation 9. The relationships are more clearly expressed by rearranging the equation to yield Equation 10. C v ( T 2−T 1 )=− p2 (V 2 −V 1) Eq. 7 C v ( T 2−T 1 )=− p2

(

) (

p2 T 1 R T2 R T 1 − =−R T 2− p2 p1 p1

(

C v ( T 2−T 1 )=− (C p−C v ) T 2− C v ( p2− p1 ) p1

=C p

(

p2 T 2 − p1 T 1

)

p2 T 1 p1

)

)

Eq. 8

Eq. 9

Eq. 10

Because the final temperature, or the temperature the gas cools to, is not known, Charles Law can be utilized to express temperature as a relationship with pressure. The initial and final temperatures can be expressed in terms of the external and final pressure, as seen below in Equation 11. Finally, the heat capacity ratio, seen in Equation 12. p2 p3 = T2 T1

Eq. 11

p1 −1 C p p2 = ᵞ= C v p1 −1 p3

Eq. 12

γ , can utilize this relationship to be expressed as

By combining CP-CV = R and the definition of

γ , we are able to obtain our working

equation to calculate CV, seen below in Equation 13. Both Equations 12 and 13 are used to calculate experimental values. C v=

R ᵞ −1

Eq. 13

For the theoretical determination of CV, we must make separate considerations for linear and non-linear molecules. These considerations are defined below:

CV =

5R 2

e x −e xi 2 i

+ R (¿¿ xi−1)2

→ For linear molecules Eq. 14

3 N−5

∑¿ i=1

e x2i −e xi CV = 3R + R (¿¿ xi−1)2

→ For non-linear molecules Eq. 15

3 N−6

∑

¿

i=1

In these equations, the first terms account for translational and rotational contributions while the second term accounts for the vibrational contributions. Xi is defined by the Equation 16 below, where h is Planck’s constant, C is the speed of light, number, k is Boltzmann’s constant, and T is temperature.

x i=

hc ṽ i kT

Eq. 16

Experimental Chemicals and Equipment

~ v i is the vibrational wave

A large carboy, a large rubber stopper, compressed gas tanks containing nitrogen, carbon dioxide, and helium, tank regulators, and a manometer filled with oil were used. Figures 1, 2, and 3 below illustrate the equipment used to complete the experiment. Carbon dioxide is a colorless gas that is present in trace amounts in our atmosphere; at low concentrations the gas is odorless [2]. Nitrogen gas comprises about eighty percent of the atmosphere and is also colorless and odorless [4]. Helium is a colorless, odorless, tasteless, and non-toxic monatomic gas that heads the group of noble gases in the periodic table [3]. All gases are potentially harmful if inhaled and can cause respiratory tract or eye irritation. Carbon dioxide can cause skin irritation and sever frostbite [2].

Figure 1. Large carboy and manometer filled with oil.

Figure 2. Compressed gas tanks with regulators containing Nitrogen, Helium, and Carbon Dioxide gas.

Figure 3. Closer view of manometer filled with oil. Experimental Procedure- See Appendix 1. Data- Raw Data

Trial

Table 1: Raw Data for Determination of the Heat Capacity of Nitrogen Gas P1 P2 P3 γ CV CV (Avg) CV (Theo) (mmHg)

1 782.15

(mmHg) 763

(mmHg) 766.59

1.23 7

(J/mol*K) 35.153

(J/mol*K)

(J/mol*K)

CV (Lit) (J/mol*K)

2

763 784.11

3

767.14 763

783.24

Trial

767.57

1

(mmHg) 763

792.38 2

(mmHg) 769.31

763 779.11

3

766.48 763

784.44

P1

32.029

20.799

767.68

1.28 4 1.28 1 1.28 7

1 785.09 784.66 784.33

(mmHg) 763

(mmHg) 771.87

763 763

774.04 772.74

1.28 5 1.25 1.26 3

20.6

27.771

CV (Lit)

(J/mol*K) 29.27

(J/mol*K)

(J/mol*K)

(J/mol*K)

29.55

29.26

25.075

28.2

28.96

Table 3: Raw Data for Determination of the Heat Capacity of Helium Gas P2 P3 γ CV CV (Avg) CV (Theo)

(mmHg)

2 3

33.162

Table 2: Raw Data for Determination of the Heat Capacity of Carbon Dioxide P1 P2 P3 γ CV CV (Avg) CV (Theo) (mmHg)

Trial

1.25 1 1.29 9

CV (Lit)

(J/mol*K) 29.183

(J/mol*K)

(J/mol*K)

(J/mol*K)

33.215 31.571

31.323

12.471

12.5

The P2 data was the same for all of the gases, and was found by searching the local weather channel to find the barometric pressure in Greenville during that particular day. All of our measured pressure values were in mmOil rather than mmHg, so we converted our values

with respect to the densities using the equation: Pi = p2 +pi(mmOil)

g /mL ( 1.48 13.6 g /mL )

, where p2

was atmospheric pressure and pi was experimental pressure, either p1 or p3, in mm of oil. This was accomplished by using an excel spreadsheet which is attached as Appendix 2. The theoretical values of Cv were calculated by taking into account the translational, vibrational, and rotational degrees of freedom for each gas. The literature values of heat capacity for each gas were obtained from an external source [5].

Data Work-Up i 0  2

Range Variable

R 8.314

J K mol

Ideal Gas Constant

Nitrogen: Data  782.15  P1N  784.11  torr    783.24 

Experimental P1 Data for adiabatic Expansion (mmHg)

766.59    P3N  767.14  torr  767.57   

Experimental P3 Data for adiabatic Expansion (mmHg)

P2N 763.00tor 

Atmospheric Pressure (mmHg)

Nitrogen: Calculations   P1N   1   P2N   N    P1N   1   P3N  

 1.237  N   1.251  1.299 CvN

   

Definition of the equation for the ratio of the heat capacity under constant pressure and constant volume

Calculated ratios of heat capacity at constant volume and pressure

R N  1



Definition of the equation for heat capacity at constant volume

35.153 

 m 2 kg

CvN 33.162 

 27.771  molKs2  

CvNavgmean( CvN)

Calculated Heat Capacity at Constant Volume Definition of the equation for calculating the mean heat capacity at constant volume.

2

CvNavg 32.029

m kg 2

molKs Navg mean(N ) Navg 1.262

Mean value of heat capacity at constant volume for Nitrogen gas. Definition of the average for heat capacity ratio value Mean value of heat capacity ratio for Nitrogen gas

Carbon Dioxide: Data  792.38  P1C  779.11 torr    784.44 

Experimental P1 Data for adiabatic Expansion (mmHg)

 769.31  P3C  766.48 torr    767.68 

Experimental P3 Data for adiabatic Expansion (mmHg)

P2C 763tor

Atmospheric Pressure

Carbon Dioxide: Calculations  P1C   1  P2C   CO2   P1C   1  P3C   The equation for the ratio of the heat capacity under constant pressure to constant volume  1.284  CO2   1.281     1.287  CvC

Calculated ratios of heat capacity at constant volume and pressure.

R CO2  1

Definition of the equation for Heat Capacity at Constant Volume

 29.27  2   m kg CvC 29.55   28.96  mol K s 2 CvCavgmean( CvC)

Calculated Heat Capacity at Constant Volume Equation for calculating the mean heat capacity at constant volume.

2

CvCavg 29.26

m kg 2

mol Ks CO2avg 1.284

Mean value of heat capacity at constant volume for carbon dioxide. Definition of Average of Heat Capacity Ratio Average of Heat Capacity Ratios

Helium Data:  785.09  P1H  784.66 torr    784.33 

Experimental P1 Data for adiabatic Expansion (mmHg)

CO2avgmean( CO2 )

 767.79  P3H  767.24 torr    767.35 

Experimental P3 Data for adiabatic Expansion (mmHg) Atmospheric Pressure

P2H 763.00tor 

Helium Calculations:   P1H    1 P2H     H    P1H   1   P3H  

   

 1.285 H   1.25  1.263  CvH

The equation for the ratio of the heat capacity under constant pressure to constant volume

Calculated ratios of heat capacity at constant volume and pressure.

R H  1

Definition of the Equation for Heat Capacity at Constant Volume

29.183    m 2 kg CvH 33.215   31.571  molKs 2   CvHavgmean( CvH)

Calculated Heat Capacity at Constant Volume Equation for calculating the mean heat capacity at constant volume.

2

CvHavg 31.323

m kg 2

molKs Havg mean(H ) Havg 1.266

Mean value of heat capacity at constnat volume for carbon dioxide. Definition of the average of the heat capacity ratios Average of the heat capacity ratios

Theoretical Calculations of Cv Using Statistical Thermodynamics i 0  2 Range Variable vnN 

2330 cm

The Vibrational Wavenumber 8 m

c 2.9979255810  34

h 6.62610

J s

s

Speed of Light Planck's Constant

 23 J

k  1.380 10

T 298.15K 

K

Boltzman's Constant Temperature

x i 

(h c vnN) k T

Equation for the calculation of the vibrational D.O.F.

Theoretical Cv for Nitrogen CvN2Theor

5 R 2

x

1

 xi 2e i

i 1

 xi  1  e 



 R

2

Equation for the calculation of theoretical heat capacity using the summation of vibrational, translational, and rotational D.O.F

2

m kg

CvN2Theor 20.799

molKs

2

Calculated Theoretical Heat Capacity

Theoretical Cv for Carbon Dioxide i 0  3 Range Variable  2349    1340   1  cm vnC  667     667  Carbon Dioxide Vibrational Wave Number Cx i 

 h c vnCi k T

Equation for the calculation of the vibrational D.O.F.

 11.341    6.469   Cx   3.22     3.22 

Calculated theoretical vib. heat capacity values for CO2

2

s K mol Cxui Cxi  2 m kg CvCO2Theor

5 R 2

Equation for adding units to CvC02 Theoretical 2



 R

i 0

Cxi

 Cx i 2 e

 Cxi  1  e 

2

Equation for the calculation of theoretical heat capacity using the summation of vibrational, translational, and rotational D.O.F

2

CvCO2Theor 25.075

m kg mol K s

2

Calculated Theoretical Heat Capacity

Theoretical Cv for Helium CvHetheor

3 R 2

Equation for theoretical heat capacity at constant volume for helium

2

m kg

CvHetheor  12.471

mol K s

2

Calculated heat capacity at constant volume for helium

Nitrogen: Propogated Error P 0.01tor

  N ( P1N P3N)   

Error in P P1N  P2N  P1N  P3N 

ddP1N ( P1N P3N) 

 

d

 1   1 

Redefining N with respect to P1N and P3N

N ( P1N P3N)

dP1N

ddP3N ( P1N P3N) 

d

Partial Derivative with Respect To P3N N ( P1N P3N)

dP3N

Partial Derivative with Respect To P1N Range Variable

i 0  2



 2 (P ) 2

ErrorinP1Ni ddP1N P1Ni P3Ni

 2.219  ErrorinP1N   2.089    3.505

10 10 10

 8  8  8

     



Error associated with P1N



ErrorinP3Ni ddP3N P1Ni P3Ni

 6.574 10 7  ErrorinP3N   5.675 10 7    7  7.16  10

Equation used to calculate the error of γ N with respect to the P1N

2

( P )

     

2

Equation used to calculate the error with the respect to the P3N

Error associated with P3N 1



2

TotalErrori  ErrorinP1N i  ErrorinP3N i

 8.244  10 4       4 TotalError  7.671  10     4   8.666  10 

Equation to calculate the total propagated error

Calculated values for the total error

Carbon Dioxide :Propogated Error  P1C    1  P2C   C ( P1C P3C)   P1C   1  P3C   ddP1C ( P1C P3C) 

d

C ( P1C P3C)

dP1C

ddP3C ( P1C P3C) 

d

Partial Derivative with Respect to P3C C ( P1C P3C)

dP3C

Partial Derivative with Respect to P1C



 2 (P ) 2

ErrorinP1Ci ddP1C P1Ci P3Ci

 1.429 10 8  ErrorinP1C   4.803 10 8    8  2.81  10

     



10

 2 (P ) 2

10 10

 7  6  7

Equation used to calculate the error of γ C with respect to the P1C

Error associated with P1C

ErrorinP3Ci ddP3C P1Ci P3Ci

 3.286   ErrorinP3C  1.063    6.158

Redefining C with respect to P1C and P3C

     

Equation used to calculate the error with the respect to the P3C

Error associated with P3C 1



2

TotalErrori  ErrorinP1C i  ErrorinP3C i

 5.856  10 4       3 TotalError  1.054  10     4   8.024  10  Helium: Propagated Error

Equation to calculate the total propagated error

Total propagated error

  H ( P1H P3H)   

P1H  P2H  P1H  P3H 

ddP1H ( P1H P3H) 

ddP3H ( P1H P3H) 

   1 

 1 

d dP1H d dP3H

Redefining γ H with respect to P1H and P3H

H ( P1H P3H)

Partial Derivative with Respect to P3H H ( P1H P3H)

Partial Derivative with Respect to P1H





ErrorinP1Hi ddP1H P1Hi P3Hi

 2.594 10   ErrorinP1H  1.974 10     2.302 10

8 8 8

2

( P )

2

     



Error associated with P1H

 2 (P ) 2

ErrorinP3Hi ddP3H P1Hi P3Hi

 5.768   ErrorinP3H  5.388    5.783

10 10 10

 7  7  7

Equation used to calculate the error of γ H with respect to the P1H

     

Equation used to calculate the error with the respect to the P3H

Error associated with P3H 1



2

TotalErrori  ErrorinP1H i  ErrorinP3H i

 7.763  10 4 ...