Lecture 1: Preliminaries Schroeder Ch. 1.1 – 1.3 Williams Ch. 1.1 – 1.4
Outline • Preliminary definitions • Temperature and thermal equilibrium • Temperature scales and thermometers • Macroscopic model of the ideal gas model • Elementary kinetic theory
What is Thermal Physics? • Thermal physics = Thermodynamics + statistical mechanics • Thermodynamics provides a framework of relating the macroscopic properties of a system to one another. • It is concerned only with macroscopic quantities and ignores the microscopic variables that characterize individual molecules • Statistical Mechanics is the bridge between the microscopic and macroscopic worlds: it links the laws of thermodynamics to the statistical behavior of molecules.
Thermodynamic Systems • A thermodynamic system is a precisely specified macroscopic region of the universe together with the physical surroundings of that region, which determine processes that are allowed to affect the interior of the region. • A thermodynamic system can be classified in three ways: – Open systems can exchange both matter and energy with the environment. – Closed systems can exchange energy but not matter with the environment. – Isolated systems can exchange neither energy nor matter with the environment.
Thermodynamic State • A thermodynamic state is the macroscopic condition of a thermodynamic system as described by a suitable set of parameters known as state variables. – Examples of state variables are temperature, pressure, density, volume, composition, and entropy. • The state variables of a given system span the thermodynamic phase space of the system and they define a space of possible equilibrium states of the system. • An essential task of classical thermodynamics is to discover a complete set of state variables for a given thermodynamics system.
Thermodynamic Processes • A thermodynamic process is any process that takes a macroscopic system from one equilibrium state to another. • In this course, we will usually examine quasi-static processes , which are sufficiently slow thermodynamic processes in which all of the state variables are well-defined along any intermediate state. – For quasi-static processes, the path in thermodynamic phase space between two states is a continuous line. • The evolution of a thermodynamic system can be given by a thermodynamic diagram . • Because there is one equation of state, all processes will occur in a two- dimensional plane, which can be spanned by any of the three possible pairs: (p,V), (p,T), and (V,T).
Temperature and Thermal Equilibrium • Consider two thermodynamic systems, A and B, that are brought into contact with one another. • Over a period of time, the net exchange of energy between both systems ceases and we say that they are in thermal equilibrium . • Thermal equilibrium is determined by a single variable called the temperature .
The Zeroth Law of Thermodynamics • Thermal equilibrium satisfies the zeroth law of thermodynamics which states – If two thermodynamic systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. • The zeroth law of thermodynamics ensures that thermal equilibrium is determined solely by temperature.
Quantifying Temperature • Q: How can we quantify temperature? • A: We can use thermometric properties to build thermometers by defining the scale of temperature in such a way that for any thermometric property 𝑌 , 𝑈 𝑌 = 𝑈 0 + 𝛽 𝑌 − 𝑌 0 We can define the two constants, 𝛽 and 𝑌 0 , that define this linear • scale by choosing two reproducible phenomenon that always occur at the same temperature. • We choose – The boiling point of pure water at sea level 𝑌 𝑐 , 𝑈 𝑐 – Triple point of pure water 𝑌 𝑔 , 𝑈 𝑔 • Using these properties, we have that 𝑔 + 𝑈 𝑐 − 𝑈 𝑔 𝑈 = 𝑈 𝑌 − 𝑌 𝑔 𝑌 𝑐 − 𝑌 𝑔
Constructing a Thermometer • A constant volume gas thermometer measures the pressure of the gas contained in the flask immersed in the bath. • The height of the mercury column tells us the pressure of the gas, and we could then find the temperature of the substance from the calibration curve. • We choose to measure two points: – The pressure of the gas when the flask is inserted into an ice-water bath and we define this as 𝑈 = 𝑈 𝑔 . – The pressure of the gas when the flask is inserted into water at the steam point and we define this as 𝑈 = 𝑈 𝑐 . • The line connecting two points on the pressure vs. temperature curve serves as a calibration curve for measuring unknown temperatures.
Temperature Scales • Choosing 𝑈 𝑔 = 0 and 𝑈 𝑐 = 100 gives the Celsius scale 100 𝑈 (𝐷) = 𝑌 − 𝑌 𝑔 𝑌 𝑐 − 𝑌 𝑔 • If we now plot the pressure vs. temperature as measured by our thermometer for different gases, we obtain a series of linear curves. • Notice that the pressure is exactly zero at 𝑈 = −273.16°𝐷 for all cases. • This is often called absolute zero and serves as the basis for a new temperature scale called the Kelvin scale . 𝑈 (𝐿) = 𝑈 (𝐷) + 273.16
Properties of Low Density Gases • The figures to the right examine the properties of low density gases while holding volume, pressure, and temperature fixed, respectively. • Examination of low density gases give the following observations: – When 𝑈 is constant, then 𝑄 ∝ 1/𝑊 ( Boyle’s law ) – When 𝑄 is constant, then 𝑊 ∝ 𝑈 ( Charles’ law ) – When 𝑊 is constant, then 𝑄 ∝ 𝑈 ( Guy- Lussac’s law ) – When 𝑈 and 𝑄 are constant, then 𝑊 ∝ 𝑜 ( Avogadro’s law ) • These relationships can be summarized: 𝑄𝑊 ∝ 𝑜𝑈
Ideal Gas Law 𝑄𝑊 ∝ 𝑜𝑈 • These observations lead to the equation of state for an ideal gas known as the ideal gas law , which is given by 𝑞𝑊 = 𝑜𝑆𝑈 • Here, 𝑜 is the number of moles in the gas and 𝑆 is called the universal gas constant with magnitude 𝑆 = 8.31 𝐾/𝑛𝑝𝑚 ∙ 𝐿 • The ideal gas law is the equation of state for an ideal gas because it gives a functional relationship between state variables 𝑄, 𝑊, 𝑈
Ideal Gas Law • The ideal gas law can be written in other forms commonly used by physicists • 1) In terms of the number of molecules 𝑶 in the gas, the ideal gas law becomes 𝑄𝑊 = 𝑂𝑙𝑈 – Here, 𝑙 is the Boltzmann’s constant with magnitude 𝐵 = 1.38 × 10 −23 𝑛 2 𝑙 𝑡 −2 𝐿 −1 𝑙 = 𝑆/𝑂 – 𝑂 𝐵 is known as Avogadro’s number • 2) In terms of the density 𝝇 of the gas, the ideal gas law becomes 𝑄 = 𝜍𝑆 ∗ 𝑈 – Here, 𝑆 ∗ is called the specific gas constant with magnitude 𝑆 ∗ = 𝑆/𝑁 – 𝑁 is defined as the molar mass of the gas in consideration (in grams).
Dalton’s Law of Partial Pressures • In a mixture of gases, each gas has a partial pressure which is the hypothetical pressure of that gas if it alone occupied the volume of the mixture at the same temperature. • For a mixture of non-reacting gases, Dalton’s law states that the total pressure exerted is equal to the sum of the partial pressures of the individual gases. • Mathematically, the pressure of a mixture of non- reactive gases can be defined by 𝑜 𝑄 𝑈𝑃𝑈𝐵𝑀 = 𝑄 𝑗 𝑗=1
Example 1 • Q: What is the specific gas constant for dry air? • A: If the pressure and density of dry air are 𝑄 𝑒 and 𝜍 𝑒 , respectively, the ideal gas equation for dry air is given by ∗ 𝑈 𝑄 𝑒 = 𝜍 𝑒 𝑆 𝑒 • Using Dalton’s law, the dry air gas constant is given by ∗ = 𝑆 𝑆 𝑒 𝑁 𝑒 • 𝑁 𝑒 is known as the apparent molecular weight of dry air
Example 1 • To determine 𝑁 𝑒 , we need to know the mole fraction of each gas constituent that comprises dry air, which is given below. • Therefore, we have ∗ = 𝑆 = 8.31 𝐾/𝑛𝑝𝑚 ∙ 𝐿 𝐾 𝑆 𝑒 ≈ 287 𝑙 𝐿 𝑁 𝑒 28.97
Example 2 • Q: What is the specific gas constant for water vapor? • A: If the pressure and density of dry air are 𝑄 𝑤 and 𝜍 𝑤 , respectively, the ideal gas equation is given by ∗ 𝑈 𝑄 𝑤 = 𝜍 𝑤 𝑆 𝑤 ∗ is the specific gas constant for water • Here, 𝑆 𝑤 vapor, given by 𝑆 𝑤 = 𝑆 = 8.31 𝐾/𝑛𝑝𝑚 ∙ 𝐿 𝐾 18.016 𝑙/𝑙𝑛𝑝𝑚 ≈ 461 𝑙 𝐿 𝑁 𝑥
The Kinetic Theory of Gases • In the previous section, we discussed the macroscopic properties of an ideal gas. • Now, we consider the ideal gas model from a microscopic point of view using kinetic theory. • The kinetic theory of gases makes the following assumptions – All molecules in the gas are identical – The molecules interact only through short-range forces during elastic collisions – The molecules obey Newton’s laws of motion – The number of molecules in the gas is large – The average separation between molecules is larger compared with their dimensions
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