1. 09: Physical properties of ideal gases
ENPh257: Thermodynamics
09: Physical properties of ideal gases
© Chris Waltham, UBC Physics & Astronomy, 2018

2. Heat capacities of ideal gases
First Law: 𝛥𝑈=𝑄−𝑊
The change in the internal energy of a system, 𝛥𝑈, equals the amount of heat added, 𝑄 , and the work done by the system, 𝑊.
Beware of the signs of these quantities, for it matters whether heat is flowing in or out, and whether work is done on or by the system or the environment.
Heat capacity, 𝐶= 𝑄/𝛥𝑇:
𝐶= 𝛥𝑈+𝑊 𝛥𝑇
𝑊= P𝛥𝑉, where P is the applied pressure and 𝛥𝑉 the change in volume (+ve 𝑊, +ve 𝛥𝑉).
© Chris Waltham, UBC Physics & Astronomy, 2018

3. Heat capacities of ideal gases
Two extremes, constant volume and constant pressure.
For constant volume, W = 0.
𝐶 𝑉 = 𝜕𝑈 𝜕𝑇 𝑉
For constant pressure:
𝐶 𝑃 = 𝜕𝑈 𝜕𝑇 𝑃 +𝑃 𝜕𝑉 𝜕𝑇 𝑃
Ideal gas law: the second term = 𝑁 𝑘 𝐵 , or 𝑛𝑅
The internal energy 𝑈= 𝑐 𝑉 𝑛𝑇 is only a function of temperature and how much matter there is, so it is irrelevant whether either 𝑉 or 𝑃 is being held constant.
N.B. Upper case 𝐶 means the heat capacity of a given volume of gas (extensive property); lower case 𝑐 means the unit heat capacity (intensive property, i.e. per mole in this case).
© Chris Waltham, UBC Physics & Astronomy, 2018

4. Constant volume, constant pressure
The heat capacity at constant volume (when the gas is doing no work on the environment) has to be less than that at constant pressure:
𝑐 𝑃 = 𝑐 𝑉 +𝑅
𝑐 𝑉 = 3 2 𝑅 for monatomic gases, i.e 𝑘𝑇 per particle per degree of freedom (= 3).
𝑐 𝑉 = 5 2 𝑅 for diatomic gases (at “normal” temperatures – too low to initiate rotation about the molecular axis), i.e 𝑘𝑇 per particle per degree of freedom (= 5).
𝑐 𝑉 =3𝑅 for more complex molecules, i.e 𝑘𝑇 per particle per degree of freedom (= 6).
Adiabatic index, 𝛾=𝑐 𝑃 / 𝑐 𝑉 , i.e. 5/3 for monatomic, 7/5 for diatomic etc.
𝑐 𝑣 𝑅
For nitrogen and oxygen, room temperature falls between Trot and Tvib.
© Chris Waltham, UBC Physics & Astronomy, 2018

5. Sound
Audible sound is the result of longitudinal waves transmitted through fluids and solids. The speed of these waves is given by: 𝑐= 𝐵 𝜌 Here B is the bulk modulus, the ratio of pressure change to the fractional change in volume: 𝐵=−𝑉 𝜕𝑃 𝜕𝑉 For an isothermal change, using the gas laws, it simply follows that 𝐵 = 𝑃 for ideal gases. However, typical periods and wavelengths of audible sound, coupled with the poor conductivity of air, make the transmission of sound in air and adiabatic process.
© Chris Waltham, UBC Physics & Astronomy, 2018

6. Sound
For adiabatic processes 𝑃 𝑉 𝛾 is constant, so 𝐵=−𝑉 𝜕𝑃 𝜕𝑉 becomes 𝐵 =𝛾𝑃. Thus the speed of sound in air is given by: 𝑐= 𝛾𝑃 𝜌 This you can show is a function of temperature alone, for a given gas. The relationship between 𝑐 and 𝑣 𝑅𝑀𝑆 is not completely trivial. 𝑐= 𝛾𝑣 𝑅𝑀𝑆 3 The factor γ comes from the adiabatic change, the 3 from the component of a random velocity in one direction.
© Chris Waltham, UBC Physics & Astronomy, 2018

7. Sound
The relationship between 𝑣 𝑅𝑀𝑆 and the temperature is 1 2 𝑚𝑣 𝑅𝑀𝑆 = 3 2 𝑘 𝐵 𝑇 𝑣 𝑅𝑀𝑆 = 3 𝑘 𝐵 𝑇 𝑚 And applying the gas laws to the expression for 𝑐: 𝑐= 𝛾 𝑘 𝐵 𝑇 𝑚 Hence: 𝑐= 𝛾 3 𝑣 𝑅𝑀𝑆
© Chris Waltham, UBC Physics & Astronomy, 2018

8. Transport properties
Armed with the kinetic theory of ideal gases, we can evaluate properties that involve heat and momentum transfer:
Conduction
Viscosity
First, we need the concept of mean free path – how far on average does a particle get before colliding with another and thus sharing energy and momentum.
For this, we need the particle size to estimate (a more precise evaluation is too complicated for this course) the mean distance between collisions.
© Chris Waltham, UBC Physics & Astronomy, 2018

9. Mean free path
Consider a sphere of diameter d, in a gas of stationary particles of number density 𝑛 𝑁 : Travelling a distance 𝑙 it will sweep out a volume 𝜋 𝑑 2 𝑙 in which any other particle will touch it. The mean distance between collisions 𝜆 is the distance travelled divided by the number of collisions: 𝜆 ~ 𝑙 𝜋 𝑑 2 𝑙 𝑛 𝑁 = 1 𝜋 𝑑 2 𝑛 𝑁 = 𝑘 𝐵 𝑇 𝜋 𝑑 2 𝑃 Here we are assuming all other molecules are stationary, but its a reasonable estimate. A more complete theory multiplies this by 1/√2. 𝑑
Number density (#/m3):
𝑛 𝑁 = 𝑁 𝐴 𝑀 𝜌= 𝑃 𝑘 𝐵 𝑇
𝑁 𝐴 = Avogadro’s Number
𝑀 = molar mass (kg)
𝜌 = mass density (kg/m3)
Beware units! Mols vs. mass, kg vs. g
© Chris Waltham, UBC Physics & Astronomy, 2018

10. Mean free path: typical numbers
Ideal gases (e.g. dry air at 1 atm, 20 C): 𝑛 𝑁 =2.5∙ per m3 Kinetic diameter d = 364 pm for N2, 346 pm for O2 Mean free path, λ = 73 nm
By Greg L at the English language Wikipedia, CC BY-SA 3.0,
© Chris Waltham, UBC Physics & Astronomy, 2018

11. Particles crossing a plane
Crude estimate - consider a box: At any one time 1/6 of the particles are heading for one face. Rate of particles passing any plane = 𝑛 𝑁 𝑣 /6 (A more complete calculation gives 𝑛 𝑁 𝑣 /4)
© Chris Waltham, UBC Physics & Astronomy, 2018

12. Thermal conduction
Conductivity 𝐾 is defined in terms of heat flow 𝑄 across an area 𝐴 under a temperature gradient 𝜕𝑇 𝜕𝑦 : 𝑄=−𝐾𝐴 𝜕𝑇 𝜕𝑦 Each molecule carries 3 2 𝑘 𝐵 𝑇 of energy. Heat flow to central plane from upper plane: 𝑛 𝑁 𝑣 𝐴 𝑘 𝐵 𝑇+ 𝜕𝑇 𝜕𝑦 𝜆 𝑦
𝑇+ 𝜕𝑇 𝜕𝑦 𝜆
𝑇+ 𝜕𝑇 𝜕𝑦 𝜆 𝜆 𝜆 𝑇 𝑇 𝜆 𝜆
𝑇− 𝜕𝑇 𝜕𝑦 𝜆
𝑇− 𝜕𝑇 𝜕𝑦 𝜆
© Chris Waltham, UBC Physics & Astronomy, 2018

13. Thermal conduction
Heat flow to central plane from upper plane: 𝑛 𝑁 𝑣 𝐴 𝑘 𝐵 𝑇+ 𝜕𝑇 𝜕𝑦 𝜆 Heat flow to central plane from lower plane: 𝑛 𝑁 𝑣 𝐴 𝑘 𝐵 𝑇− 𝜕𝑇 𝜕𝑦 𝜆 𝑦
𝑇+ 𝜕𝑇 𝜕𝑦 𝜆 𝜆 𝑇 𝜆
𝑇− 𝜕𝑇 𝜕𝑦 𝜆
© Chris Waltham, UBC Physics & Astronomy, 2018

14. Thermal conduction
Net heat flow: 3 𝑛 𝑁 𝑣 𝐴 4 𝑘 𝐵 𝜕𝑇 𝜕𝑦 𝜆 Identify conductivity of an ideal gas to be: 𝐾= 3𝑛 𝑁 𝑣 4 𝑘 𝐵 𝜆= 3 𝑣 λ 4 𝑃 𝑇 Recall λ α 𝑇 𝑃 so λ𝑃 𝑇 is a constant for a given gas, so thermal conductivity of an ideal gas should be proportional to velocity, (i.e. the square root of temperature) and independent of pressure. 𝑦
𝑇+ 𝜕𝑇 𝜕𝑦 𝜆
𝑇+ 𝜕𝑇 𝜕𝑦 𝜆 𝜆 𝜆 𝑇 𝑇 𝜆 𝜆
𝑇− 𝜕𝑇 𝜕𝑦 𝜆
𝑇− 𝜕𝑇 𝜕𝑦 𝜆
Calculation for air at 20 C and 1 atm yields W/m⋅K, compared to the measured value of
© Chris Waltham, UBC Physics & Astronomy, 2018