Determining the Specific Heat Capacity of Air

Contents

Ⅰ ： Aim

Ⅱ ： Introduction

Ⅲ ： Theory

Ⅳ ： Experimental Process

Ⅴ ： Instruments and Data Table

Ⅰ： Aim

• To measure the specific heat ratio of air by the method of adiabatic expansion.

• To learn how to use the temperature sensor and the pressure sensor.

Ⅱ： Introduction• The heat capacity ratio or adiabatic index or ratio of

specific heats, is the ratio of the heat capacity at constant pressure (CP) to heat capacity at constant volume (CV). It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma).

where, C is the heat capacity or the specific heat

capacity of a gas, suffix P and V refer to constant pressure and constant volume conditions respectively.

V

P

C

C

Ideal gas relations

• For an ideal gas, the heat capacity is constant with temperature. Accordingly we can express the enthalpy as H = CPT and the internal energy as U = CVT. Thus, it can also be said that the heat capacity ratio is the ratio between the enthalpy to the internal energy:

Ideal gas relations

• Furthermore, the heat capacities can be expressed in terms of heat capacity ratio ( γ ) and the gas constant ( R ):

and

So ：

1

R

CP1

RCV

RCC VP

Relation with degrees of freedom

• The heat capacity ratio ( γ ) for an ideal gas can be related to the degrees of freedom ( f ) of a molecule by:

• Thus we observe that for a monatomic gas, with three degrees of freedom:

• while for a diatomic gas, with five degrees of freedom (at room temperature):

f

f 2

67.13

5

4.15

7

E.g.

• The terrestrial air is primarily made up of diatomic gasses (~78% nitrogen (N2) and ~21% oxygen (O2)) and, at standard conditions it can be considered to be an ideal gas. A diatomic molecule has five degrees of freedom (three translational and two rotational degrees of freedom). This results in a value of

4.15

7

Ratio of Specific Heats for some common gases

Gas Ratio of Specific Heats

Carbon Dioxide 1.3

Helium 1.66

Hydrogen 1.41

Methane or Natural Gas 1.31

Nitrogen 1.4

Oxygen 1.4

Standard Air 1.4

One Standard Atmosphere

Common Pressure Units frequently used as alternative to "one Atmosphere"

• 76 Centimeters (760 mm) of Mercury • 10.332 Meters of Water • 101.33 Kilopascal Note: Standard atmosphere is a pressure defined as

101'325 Pa and used as unit of pressure (symbol: atm).

The original definition of “Standard Temperature and Pressure” (STP) was a reference temperature of 0 °C (273.15 K) and pressure of 101.325 kPa (1 atm).

Ⅲ： Theory• Ideal gas law

– The state of an amount of gas is determined by its pressure, volume, and temperature according to the equation:

where

P is the absolute pressure of the gas,V is the volume of the gas,n is the number of moles of gas,R is the universal gas constant,T is the absolute temperature.

The value of the ideal gas constant, R, is found to be as follows. R = 8.314472J·mol−1·K−1

nRTPV

Calculations

Process Constant EquationIsobaric process Pressure V/T=constant

Isochoric process Volume P/T=constant

Isothermal process

Temperature PV=constant

Isentropic process

(Reversible adiabatic process)

Entropy

PVγ=constant

Pγ-1/Tγ=constant

TVγ-1=constant

Isotherms of an ideal gas

T: high

T: low

Ⅳ: Experimental process

Ⅰ(P1,T0) Ⅱ(P0,T1) Ⅲ(P2,T0)

Adiabatic expansion

Isochoric process (pressure increase)

Calculations

1

0

1

0

1

1

10

0

11

1

T

T

P

P

T

P

T

P

constT

P

Adiabatic expansion. (Ⅰ P1,T0)---- (Ⅱ P0,T1)

Equation 1

Calculations

Isochoric process (pressure increase).

Ⅱ(P0,T1)------ (Ⅲ P2,T0)

0

1

2

0

0

2

1

0

T

T

P

P

T

P

T

P

constT

P

Equation 2

Calculations

Through equation 1 and 2

2

1

0

1

0

2

0

1

0

2

1

0

1

ln

ln

lnln1

PP

PP

P

P

P

P

P

P

P

P

Equation 3

Ⅴ:Instruments and data table

Testing Instrument

Sensitivity

• The Pressure Sensor:20mV/kPa

• The Temperature Sensor:5mV/K

)(200 a

ii kP

PPP

)(5

KU

T T

Data table

P0 （ kPa）

T0

ΔP1

(mV )

P1

(kPa)

ΔP1

(mV)

P2

（ kP

a）γ

1

101.30

2

3

4

5

6

Result

0 1.402

0

0

100%E