Lecture 37 - Thermal Physics II PDF

Lecture 37 – THERMAL PHYSICS II Introduction to Thermodynamics Thermodynamics is the study of heat and how it flows. This lecture will cover the laws of thermodynamics, which are essential to physics. Despite the relatively recent discovery of atoms and molecules a century ago, the principles of the...

Lecture 37 – THERMAL PHYSICS II Introduction to Thermodynamics Thermodynamics is the study of heat and how it flows. This lecture will cover the laws of thermodynamics, which are essential to physics. Despite the relatively recent discovery of atoms and molecules a century ago, the principles of thermodynamics remain unchanged and deeply rooted in our understanding of matter. Interestingly, we now interpret heat as the movement of atoms and molecules. As previously discussed, exerting work on an object increases its temperature due to the accelerated, random motion of its particles (atoms and molecules). Notably, the foundational thermodynamic laws were established without knowledge of these microscopic constituents, yet they still yield accurate predictions. System: The specific part of the universe being studied, which can be a quantity of matter or a region in space. Surroundings: Everything outside the system that can interact with it. State Variables: The quantities that define the state of a thermodynamic system at equilibrium depending only on the current state of the system, not on how the system reached that state. Examples include pressure (P), volume (V), temperature (T), and internal energy (U). Thermodynamic Variables: This is a broader term that encompasses state variables and may include other quantities used in thermodynamic analyses, such as heat (Q) and work (W). Unlike state variables, heat and work are path-dependent and not properties of the system's state. Equation of State: The mathematical relation between these variables. Consider a gas with N atoms or molecules. The gas has a volume V, pressure P, and temperature T. These are thermodynamic variables but not independent of each other. For example, in a container with N molecules, if the volume increases, the pressure decreases (Boyle's law: P is inversely proportional to V) with constant temperature. Alternatively, under constant pressure, if the temperature increases, the volume increases (Charles's law: V is proportional to T). These relationships form the equation of state, where pressure is a function of volume and temperature, and vice versa. P = f (V, T) V = g (P, T) T = h (P, V) The equation of state for an ideal gas is given by:  PV = NkBT where kB is Boltzmann's constant, T is the absolute temperature measured in Kelvin (Kelvin = Celsius + 273.15). Ideal Gas: An ideal gas consists of widely spaced atoms or molecules that rarely collide with each other, allowing for mostly free movement. They do collide with the container walls, but inter-particle collisions are minimal. This concept is crucial for understanding thermodynamic relationships. Internal Energy: It is the average kinetic energy of a system's atoms and molecules (fig 37.1), which increases with temperature. It can be expressed mathematically, 3 E= NkBT 2 Figure 37. 1. Random motion of gas molecules contributing to the internal energy of the system. First Law of Thermodynamics The first law of thermodynamics—a statement of energy conservation— states that when work is done on a body and it is heated, the internal energy of the system increases. Mathematically, the increase in internal energy (ΔE) equals the heat added (ΔQ) plus the work done (ΔW), expressed as ΔE = ΔQ + ΔW ΔQ and ΔW are positive if heat is added to the system or work is done on a system. ΔQ and ΔW are negative if heat is removed from the system or work is done by a system. This law highlights that while internal energy does not depend on the path but work and heat are path variables – their values depend on the steps leading from one state to another. Consider a closed system with a piston, shown in fig 37.2: it's sealed except for a movable piston, containing an ideal gas. When the piston moves, it alters the system's volume without allowing gas to escape, demonstrating the relationship between work, heat, and internal energy in a controlled environment.  Figure 37. 2. Closed system with a movable piston, changing the volume of an ideal. To determine the work done by a gas during expansion, If the external pressure is Pexternal, the work done by the gas on the piston, dW, is given by: dw = - Pexternal. dV ⸫ w=F.d=(PA)d=P(Ad)=PV where dV is the change in volume. The negative sign indicates that the work is done by the gas on the surroundings. When internal and external pressures are equal, the piston is in equilibrium; If Pinternal > Pexternal, the gas expands, and If Pinternal < Pexternal, it compresses. For total work, we integrate Pexternal.dV from the initial to the final volume. V2 w = −  Pext dV V1 Case I: Pexternal = 0 (a vacuum) No work is done by the gas on the piston as it expands, since work is a product of force and displacement, and with zero external pressure, there's no force opposing the expansion. Consequently, if no heat is exchanged (Q = 0), the system's internal energy remains unchanged (so, E = 0). This is consistent with the first law of thermodynamics, which relates changes in internal energy to heat and work. Case II: Pexternal = constant With constant external pressure Pexternal, the work done by the gas during expansion is given by W = - Pexternal.V where V is the change in volume (i.e., Vfinal - Vinitial). The negative sign signifies that the gas is doing work on the surroundings (gas expands against the external pressure). Case II: Pexternal ≈ Pinternal Assume Pexternal and Pinternal to be the same for calculation purposes. Using the ideal gas law (PV = NkT), the work done during an isothermal process (constant temperature) is calculated by integrating the pressure with respect to volume NkT Pext = V NkT V2 dV V = − NkT ( ln V2 − ln V1 ) = − NkT ln 2 V2 w = − dV = − NkT  V1 V V1 V V1 where V1 and V2 are the initial and final volumes, respectively. These cases illustrate that the work done depends on the path taken to reach a particular state of the gas: 1. Free expansion: No work is done (w = 0). 2. Constant external pressure: Work is - Pexternal V. V2 3. Isothermal Process with Pexternal ≈ Pinternal: Maximum work is w = − NkT ln. V1 Thus, the amount of work is path-dependent, and asking for the work done by the gas without specifying the process is incorrect. The internal energy, however, is a state function and remains the same regardless of the path taken. When a valve separating a gas from a vacuum is opened, the gas expands without doing work, as there is no external pressure to work against. Since it is an isolated system undergoing free expansion, no external work is done (W = 0) and no heat is exchanged (Q = 0), so the internal energy remains constant (E = Q + W = 0). Initially, if the energy is 3/2NkT, it stays the same after expansion. PV diagrams PV diagrams are helpful in visualizing this process, where the area under the curve represents the work done. For a constant pressure process (see fig. 37.3), the work done is the product of the pressure and the change in volume (W = P V).  Figure 37. 3. PV diagram for constant pressure process. Path-Dependance of Work The work done can be different depending on the path taken. For example, if we move from an initial state P1V1 to a final state P2V2 via two different paths, the work done will vary. In an isothermal expansion, where temperature remains constant, the work done is determined as follows: Path A: Firstly, the volume increases from V1 to V2 at constant pressure P1. Then, the pressure decreases to P2 at constant volume V2, shown in fig 37.4 (path A). Since the temperature remains constant, there's no change in internal energy. Thus, the work done is pressure (starting pressure) times the change in volume. W = P1(V2 - V1) Figure 37. 4. PV diagrams showing path dependence of work Path B: Exploring alternative paths (Path B in fig 37.4) in thermodynamics demonstrates the path-dependence of work done on or by a gas. For instance, on path B, First, decrease the pressure from P1 to P2 at constant volume V1. Then, increase the volume from V1 to V2 at constant pressure P2., the work done is W = P2 (V2 − V1 ) This contrasts with the previous path where the work was W = P1 (V2 − V1 ).This shows that work done is indeed path-dependent, and the signs of work done on or by the gas are opposite. Path C: For the third path (fig 37.5), which adheres to the ideal gas law PV = NkT, the work done is calculated as: For an isothermal process, PV = Nk BT dW = PdV T = Nk B dV V V2 V2 Nk BT W =  PdV =  dV V1 V1 V V2 W = Nk BT ln V1 Figure 37. 5. Path C showing variable change in pressure and volume. Specific Heat Now, let’s calculate the heat capacity of a gas, it's a measure of the amount of heat required to change the temperature of a system. There are two ways to define specific heat: constant volume (CV) and constant pressure (CP). Specific Heat at Constant Volume It (CV) is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin) while keeping its volume constant. For an ideal gas at constant volume (CV), QV CV = T QV = CV T As the volume remain constant, dV = 0, Hence, W =0 QV = E + W = E E = CV T As T → 0 dE = CV dT dE 3 CV =  = NkT dT 2 d 3  3 CV =  NkT  = Nk dT  2  2 This is a fundamental result in thermodynamics, reflecting the intrinsic energy changes within a gas due to temperature variations. Specific Heat at Constant Pressure CP represents the heat capacity of a gas when it is allowed to expand under constant pressure. This means the system can do work on its surroundings as it heats up. CP is defined as the heat absorbed QP divided by the change in temperature ΔT. QP CP = T QP = CP T Let’s calculate the specific heat at constant pressure CP using the first law of thermodynamics, QP = E + W = E + PV CP T = E + PV As T → 0 CP dT = dE + PdV Expressing dE in terms of CV CP dT = CV dT + PdV As PV = Nk BT Differentialing w.r.t T PdV + VdP = Nk B dT At const P (dP=0) PdV = Nk B dT Substitute into the first law CP dT = CV dT + Nk B dT CP = CV + Nk B 3 E= Nk BT 2 dE 3 CV = = Nk B dT 2 3 CP = Nk B + Nk B 2 5 CP = Nk B 2 5 3 CP − CV = Nk B − Nk B = Nk B 2 2 This shows that CP is indeed greater than CV for an ideal gas, by an amount equal to NkB, which is the number of particles multiplied by Boltzmann's constant. The difference arises because at constant pressure, the gas does work on the surroundings as it expands, requiring additional energy input to raise the temperature by the same amount compared to a constant volume scenario. The behavior of an ideal gas can become complex when considering real molecules that have rotational and vibrational modes. For diatomic or polyatomic molecules, such as water (H₂O), carbon dioxide (CO₂), or hydrogen (H₂), their rotational motions contribute to their energy and, consequently, to their specific heat capacities displayed in fig 37.6. The specific heat at constant volume CV for a monatomic gas is 3/2 nK, accounting for translational motion along the x, y, and z axes. For diatomic or polyatomic gases, which can rotate, additional degrees of freedom contribute to the specific heat. For example, a diatomic molecule such as CO2 has two additional rotational degrees of freedom. Figure 37. 6. Diatomic molecule having translational and rotational degrees of 1 2 1 2 1 2 1 1 K= mv x + mv y + mv z + I xx2 + I y y2 freedom. 2 2 2 2 2 1  5 E = 5   Nk BT  = Nk BT 2  2 5 CV = Nk B 2 Therefore, molecules that can rotate have a higher specific heat. Adiabatic Process So far, we've discussed gas expansions under isothermal (constant temperature) and constant pressure conditions. The third type of expansion is adiabatic, where no heat is exchanged with the surroundings (dQ = 0). An isolated system executing adiabatic expansion is shown in fig 37.7. dQ = dE + dW = 0 or dE = −dW This means any work done by the gas reduces its internal energy. For an adiabatic process, the relationship between pressure and Figure 37. 7. An isolated volume is given by system in which no heat enters or leave. CV dT + PdV = 0 dV Nk BT CV dT + Nk BT =0 P = V V dT Nk B dV + = 0................(1) T CV V  As CP − CV = Nk B Nk B CP − CV CP = = −1 CV CV CV CP Define  CV Nk B =  − 1, put in eq (1) and integrate CV dT dV + (  − 1) =0 T V ln T + (  − 1) ln V = Constant ln (TV  −1 ) = Constant TV  −1 = Constant PV  −1 PV V = Constant  PV =Nk BT → T = NkB Nk B PV  = Constant For an ideal gas undergoing an adiabatic expansion from an initial state P1V1 to a final state P2V2, the work done is: dQ = dE + dW = 0 CV dT + dW = 0  dW = −CV dT Wadiabatic = dW = −  CV dT = −CV T Wadiabatic = −CV T = CV (T1 − T2 )  PV PV  PV − PV Wadiabatic = CV  1 1 − 2 2  = 1 1 2 2  Nk B Nk B   −1 This is a direct application of the first law of thermodynamics, which is fundamentally the law of conservation of energy. It states that the change in internal energy of a system is equal to the heat added to the system plus the work done on the system. Second Law of Thermodynamics The second law of thermodynamics asserts that there can be no process whose only final result is to transfer thermal energy from a cooler object to a hotter object. This principle implies that it's impossible to extract work from a system solely by transferring heat from a colder to a hotter body. Applications of Second Law of Thermodynamics This principle has numerous applications, particularly in engines. Fig 37.8 shows a steam engine in which water is heated in a boiler until it becomes steam. This steam then expands and pushes a piston, which in turn drives a wheel, converting the steam's thermal energy into mechanical work. After doing work, the steam is condensed back into water, which can be heated again, creating a cycle.  Figure 37. 8. Steam engine Similarly, internal combustion engines (fig 37.9) in motorcycles and cars function on the same basic principle. Gas enters the engine, is compressed, and ignited with a spark (ignition stroke). The piston then moves out (power stroke) and finally, the exhaust stroke occurs. Figure 37. 9. Internal combustion engine in vehicles demonstrating (a) compression stroke (b) ignition stroke (c) power stroke (d) exhaust stroke These engines operate between a high temperature (over 100°C for steam, higher for diesel or petrol during ignition) and a low temperature (30-40°C for exhaust release). Heat, Qh, is taken from the high temperature reservoir Th, allowing the engine to perform work. Not all heat converts to work; some is rejected as excess heat, Qc, to the environment, called the cold reservoir at temperature Tc. Fig 37.10 demonstrates the schematics of heat engine.  Figure 37. 10. Working mechanism of heat engine. This follows the first law of thermodynamics: Q = E + W W = Qin ,h − Qout ,c Efficiency (ε) measures the work obtained from heat input: mathematically defined as the work output (W) divided by the heat input (Qin). W Q − Qout ,c Q = = in ,h = 1 − out ,c Qin ,h Qin ,h Qin ,h Since Qout, c is proportional to the lower temperature (Tc) and Qin, h to the higher temperature (TH), efficiency is: TC  = 1− Th This ensures efficiency is always less than 1, making 100% efficiency impossible. Refrigerator A refrigerator operates in reverse compared to a heat engine. While a heat engine converts heat from a high temperature to work and expels some waste heat, a refrigerator removes heat from a low temperature area (like a water bottle inside it) and rejects it at a higher temperature. This process requires external energy, usually electricity. In a refrigerator, work (W) is input, and heat (Qc) is removed from inside and rejected as Qh outside (fig 37.11). The efficiency of a refrigerator is measured by its coefficient of performance, which is the amount of heat extracted per unit of work input. The second law of thermodynamics states that a refrigerator cannot operate without external work; it is impossible for a refrigerator to transfer heat from a cold to a hot body without additional energy input. Thus, its efficiency will always be less than 1. Entropy Entropy, a concept from the second law of thermodynamics, measures disorder in Figure 37. 11. Working mechanism of refrigerator. a system. Mathematically, if you add a small amount of heat (dQ) to a system at temperature (T), the increase in entropy (dS) is: dQ ds = T Entropy of an ideal gas For an ideal gas, the state equation is PV = nKT. When heat is added, the entropy change can be derived from the first law of thermodynamics: dQ = dE + dW = dE + PdV Substituting P from the ideal gas law, dV dQ = CV dT + Nk BT V dQ dT dV dS = = CV + Nk B T T V To find the total change in entropy ΔS when the temperature changes from T1 to T2, we integrate: dQ T V S =  =CV ln 2 + NkB ln 2 T T1 V1 Physically, as energy is added to a system and temperature increases, molecules move faster (more randomly), increasing disorder and entropy. More order means less entropy, and more disorder means higher entropy. The second law of thermodynamics states that entropy always increases in an isolated system. Over time, any isolated system, whether gas, liquid, or solid, will naturally become more disordered. For example, imagine gathering all the atoms and molecules scattered in a room and confining them in small box. This action decreases entropy because the system becomes more ordered. However, if you open the box and allow the atoms and molecules to disperse throughout the room, entropy increases due to the rise in disorder. This concept also links entropy to the flow of time. We never observe atoms and molecules spontaneously gathering in one place, which would decrease entropy. Thus, the increase in entropy is a measure of time's forward progression.