Applications of the First Law Chapter 3

1. Applications of the First LawChapter 3 CHEM 321-01 Dr. Daniel E. Autrey Fayetteville State University

2. State and Path Functions • State function • Properties that are independent of how a state is prepared. • Functions of variables that define the current state of a system, such as pressure, temperature, and volume. • Internal energy (U) and enthalpy (H) are state functions. • Properties that the state “possesses.” • Path function • Properties that relate to the path or preparation of the state of the substance. • The work (w) done in preparing the state and the energy transferred as heat (q) are path functions. • Since systems do not possess work or heat, but perform or transfer them, they are not properties of the current state of the system, but refer to the path taken by the system in attaining the current state.

3. State and Path Functions • As the volume and temperature of a system are changed, the internal energy changes. Consider the illustration: • The initial state has an internal energy, Ui. • The final state has an internal energy, Uf. • The final state can be achieved by two possible paths: • Path 1: adiabatic expansion, q =0.The ΔU is equal to the work (w) done on the system. • Path 2: non-adiabatic expansion, w' and q' are both done on the system. • ΔU is the same, regardless of the path. • Work (w) and heat (q) depend on the path taken between the states.

4. Exact Differentials • An infinitesimal change in the internal energy of a system is represented as the differential, dU. • For a complete process, dU is integrated from the initial to the final conditions. • Mathematically, • When dU is integrated, the result is the change in internal energy for the process. • dU is an exact differential because its integrated value of ΔU is path-independent. • All state function have exact differentials.

5. Inexact Differentials • Infinitesimal changes in the work or heat of a system are represented as the differentials, dw and dq. • For a complete process, dw and dq are integrated from the initial to the final conditions. • Mathematically, • Whendw and dq are integrated, the result is the absolute amount of work and heat for the process. • The differentials dw and dq are called inexact differentials because their integrated values w and q are path-dependent.

6. Euler Chain Rule • The Euler Chain Rule is a very useful relation between partial derivatives. • For a differentiable function z : • The product of the three partial derivatives is:

7. Changes in Internal Energy • Because the internal energy (U) of a system is a state function, it may be expressed as a function of the physical properties of the system, such as the pressure (p), volume (V), and temperature (T). • If the system happens to be a closed system, in which the amount (n) of gas remains constant, the internal energy may conveniently be expressed as a function of the volume and temperature. • Because the pressure is related to the volume and temperature by an equation of state (such as the ideal gas law), it is not an independent variable.

8. Changes in Internal Energy • For a closed system at constant composition (n), the internal energy (U) of a system is a function of the volume (V) and temperature (T): U(V, T) • When the volume changes infinitesimally from V to V+dV at constant temperature (T), the internal energy changes from its initial state (Ui) to its final state (Uf):

9. Changes in Internal Energy • Similarly, when the temperature changes infinitesimally from T to T+dT at constant volume (V), the internal energy changes from its initial state (Ui) to its final state (Uf):

10. Changes in Internal Energy • If both the temperature and volume change infinitesimal amounts, dT and dV, the internal energy changes from its initial state (Ui) to its final state (Uf):

11. Changes in Internal Energy • Since the change in internal energy is infinitesimal, we can express the change between the initial and final states as the exact differential dU: • The significance of this equation is that, in a closed system of constant composition (n), any infinitesimal change in the internal energy (U) is proportional to the infinitesimal changes of volume (V) and temperature (T).

12. Changes in Internal Energy • Recall that the heat capacity at constant volume, CV, is defined as: • The heat capacity at constant volume, CV, is the slope of the internal energy (U) with respect to the temperature (T) and constant volume (V).

13. Changes in Internal Energy • The internal pressure (πT) is the measure of the change in the internal energy of a substance as its volume is changed at constant temperature. • Mathematically, the internal pressure (πT) is defined as:

14. Changes in Internal Energy • Because an infinitesimal change in the internal energy (dU) is related to a infinitesimal changes in volume (dV) and temperature (dT): • Substituting: • Gives:

15. Internal Pressure (πT) • The internal pressure (πT) is a measure of the cohesive forces in the sample. • Recall that: • For a perfect gas, in which there are no interactions between the particles, the internal energy (U) is independent of the separation between the particles, and thus independent of the volume (V) of the sample. • As a result, the internal energy (U) is independent of the volume (V) of the sample at constant temperature (T) . • For an ideal gas:

16. Internal Pressure (πT) For Real Gases • Real gases can have either attractive or repulsive forces dominating. • When attractive forces between the particles dominate, the internal energy (U) increases as the sample expands isothermally because the molecules become further apart on average than they would like. In this case the internal pressure is πT > 0, and dU > 0. • When repulsive forces between the particles dominate, the internal energy decreases (U) as the sample expands isothermally. In this case the internal pressure is πT < 0, and dU < 0.

17. The Joule Experiment • James Joule attempted to measure the internal pressure (πT) by observing the change in temperature (ΔT) of a gas when it is allowed to expand in a vacuum. • He placed two metal vessels joined together by a stopcock in a water bath at constant temperature in which one vessel was filled with air at about 22 atm and the other was evacuated. • When the gas expanded, he observed no temperature change (ΔT = 0).

18. The Joule Experiment • Why? • First, the gas experienced a free expansion, so the system did no work, (w = 0). • Second, because the bath was at constant temperature, no heat was able to enter or leave the system, and the system was adiabatic (q = 0). • Because the system was an adiabatic free expansion, ΔU = 0. • Joule’s crude experiment showed that the internal energy (ΔU) did not change much when a gas expands isothermally. Therefore, πT = 0. • The Joule experiment extracted a limiting property of an ideal gas because the heat capacity of the apparatus was so large that the actual change in temperature caused by the isothermal expansion of a real gas was too small to measure.

19. Changes in Internal Energy at Constant Pressure • How does the internal energy (U) vary with temperature (T) at constant pressure (p)? • The infinitesimal change in internal energy (dU) is related to infinitesimal changes in volume (dV) and temperature (dT) by: • Dividing both sides by dT and imposing the constraint of constant pressure:

20. Changes in Internal Energy at Constant Pressure • The variation of internal energy (U) with temperature (T) at constant pressure (p) is: • The expansion coefficient (α) of a substance is related to the variation of volume (V) with temperature (T) at constant pressure (p) and is defined as:

21. Changes in Internal Energy at Constant Pressure • Substituting the expression for the expansion coefficient (α) gives: • Since πT = 0 for a perfect gas, • Thus, for a perfect gas:

22. Expansion Coefficient For a Perfect Gas • The expansion coefficient (α) for a perfect gas is: • Substituting the expression for volume (V) from the ideal gas law:

23. Expansion Coefficient For a Perfect Gas • Taking the derivative with respect to temperature at constant pressure: • The expansion coefficient (α) is inversely proportional to the temperature for a perfect gas. • The higher the temperature, the smaller the expansion coefficient, and the volume is less responsive to a change in temperature.

24. Changes in Enthalpy • The enthalpy (H) is defined as: • Because the internal energy (U), pressure (p) , and volume (V) are all state functions, the enthalpy (H) is also a state function, and the differential (dH) is exact. • The enthalpy (H) may be expressed as a function of the state variables: p, T, and V. If the system happens to be a closed system, in which the amount (n) of gas remains constant, the enthalpy may conveniently be expressed as a function of the pressure and temperature. H(p, T)

25. Changes in Enthalpy • An infinitesimal change in the enthalpy (dH) is related to a infinitesimal changes in pressure (dp) and temperature (dT): • Recall that the expression for the heat capacity at constant pressure, Cp : • Substituting the expression for Cp gives:

26. Changes in Enthalpy at Constant Volume • Dividing both sides by dT and imposing the constraint of constant pressure: • From the Euler Chain Rule: • Rearranging:

27. Changes in Enthalpy at Constant Volume • Applying the reciprocal identity: • Recall that the expansion coefficient (α): • Substituting:

28. Changes in Enthalpy at Constant Volume • The isothermal compressibility (κT) of a gas is defined as: • The negative sign ensures that κT > 0 because an increase in pressure (positive dp) causes a reduction of volume (a negative dV).

29. Changes in Enthalpy at Constant Volume • Therefore, the derivative: • Substituting into the expression: • Gives:

30. Changes in Enthalpy at Constant Volume • Recall that: • Substituting into the expression: • Gives:

31. Changes in Enthalpy at Constant Volume • What is the significance of • Applying the Euler Chain Rule: • Rearranging:

32. Changes in Enthalpy at Constant Volume • Applying the reciprocal identity: • Recall that the expression for the heat capacity at constant pressure, Cp : • Substituting the expression for Cp gives:

33. Changes in Enthalpy at Constant Volume • The Joule-Thomson coefficient (μJT) is defined as: • Substituting the expression for μJT into: • Gives:

34. Changes in Enthalpy at Constant Volume • Recall that: • Substituting: • Gives:

35. Isothermal Compressibility (κT) For a Perfect Gas • The volume of a perfect gas is: • Taking the partial derivative of volume with respect to pressure at constant temperature:

36. The higher the p, the less compressible. Isothermal Compressibility (κT) For a Perfect Gas • The isothermal compressibility (κT) of a perfect gas: • Substituting: • Simplifying:

37. The Joule-Thomson Effect • The analysis of the Joule-Thomson coefficient (μJT) is helps to understand how gases are liquefied. • James Joule and William Thomson (later Lord Kelvin) developed an apparatus for measuring: • Measuring the Joule-Thomson coefficient (μJT) requires the constraint of constant enthalpy, or an isenthalpic process.

38. The Joule-Thomson Effect • The apparatus for measuring the Joule-Thomson coefficient (μJT): • The process: • Continuous stream of gas at a high pressure is allowed to expand into a region of low pressure across a porous cotton plug. • The apparatus was insulated from the surroundings, so the process is adiabatic. • The difference in temperature resulting from the expansion is monitored. • What was observed? • Lower temperature on the low pressure side. • The difference in temperature was proportional to the difference in pressure.

39. The Joule-Thomson Effect • How was constant enthalpy maintained? • The process: • The process is adiabatic, so q = 0. Thus, ΔU = w. • The gas on the high-pressure side has a pressure pi and temperature Ti, and occupies a volume ofVi. • The gas on the low-pressure side has a pressure pf and temperature Tf, and occupies a volume ofVf. • The flow of gas acts as “pistons.” The upstream pressure isothermally “compresses” the gas, simultaneously as the downstream pressure isothermally “expands” the gas.

40. The Joule-Thomson Effect • The gas on the high-pressure side is isothermally compressed against a constant pressure pi until the volume Vi = 0. The work done on the gas is: • The gas on the low-pressure side isothermally expands against a constant pressure pf from a volume V = 0→ Vf. The work done by the gas is:

41. isenthalpic condition The Joule-Thomson Effect • Since work is a path function, the total work done on gas is: • Because the process is adiabatic, the change in internal energy of the expanding gas is: • Separating the initial and final conditions: • Gives (definition of enthalpy):

42. The Joule-Thomson Coefficient • The property measured in the Joule-Thomson Experiment is: • Adding the constraint of constant enthalpy, and taking the limit of an infinitesimal change in pressure, Δp → 0: • The Joule-Thomson coefficient (μJT) is the ratio of the change in temperature to the change in pressure when a gas undergoes isenthalpic adiabatic expansion.

43. The Joule-Thomson Coefficient • From the Euler Chain Rule: • Solving for the Joule-Thomson coefficient: • From the reciprocal identity:

44. μJT may be determined from Cp and α The Joule-Thomson Coefficient • A relationship (to be derived later) is: • Substituting into the expression for the Joule-Thomson coefficient: • Gives:

45. The Joule-Thomson Coefficient • The Joule-Thomson coefficient may be estimated from the van der Waals constants: • The van der Waals equation also gives an explanation for the existence of an inversion temperature. • The van der waals parametera is a measure of the attractive forces between molecules. • For gases with strong attractive forces, a is large. • For gases with negligible attractive forces, a is very small.

46. The Joule-Thomson Coefficient • For gases with appreciable intermolecular forces: and μJT > 0. • For hydrogen and helium gases at ambient temperatures, a is quite small. Thus, and μJT < 0. • These gases have to be pre-cooled to make μJT positive.

47. Isothermal Joule-Thomson Coefficient • The isothermal Joule-Thomson coefficient (μT) is: • It is the slope of enthalpy against pressure at constant temperature. • The isothermal Joule-Thomson coefficient is related to the Joule-Thomson coefficient by:

48. Isothermal Joule-Thomson Coefficient • The isothermal Joule-Thomson coefficient (μT) is measured indirectly. • The gas at a steady pressure is pumped through a heat exchanger, which sets the temperature. • The gas then passes through a porous plug inside a thermally insulated container, making the process adiabatic. • The steep pressure drop (Δp) is measured. • The cooling effect is exactly offset by an electric heater placed immediately after the plug. • The energy (heat) provided by the heater is monitored, which equals ΔH, because the pressure is essentially constant after the porous plug. • In the limit as Δp → 0, the isothermal Joule-Thomson coefficient is calculated by:

49. Liquefaction of Gases • The Joule-Thomson coefficient is also a measure of the nonideality of gases. • For an ideal gas, μJT = 0, and the temperature does not drop when an ideal gas undergoes Joule-Thomson expansion. • The degree of cooling experienced by a real gas undergoing isenthalpic adiabatic expansion is related to the degree of intermolecular forces in that gas. • Real gases have nonzero Joule-Thomson coefficients. The sign of μJT depends on: • The identity of the gas. • The pressure. • The relative magnitudes of attractive and repulsive forces. • The temperature.

50. Liquefaction of Gases • If μJT > 0, the gas cools when it adiabatically expands. This occurs when dT is negative and dp is negative. • If μJT < 0, the gas heats up when it adiabatically expands. This occurs when dT is positive and dp is negative. • An inversion temperature is the temperature at which the Joule-Thomson coefficient changes signs. • Gases that show a heating effect at one temperature will show a cooling effect when the gas is cooled beneath their inversion temperature.