Chemistry Lecture 10 Maxwell Relations NC State University
Thermodynamic state functions expressed in differential form We have seen that the internal energy is conserved and depends on mechanical (dw) and thermal (dq) energy. Given that pressure-volume work done by the system is dw = -PdV and that the entropy change can be expressed as ds = dq/t (i.e. dq = TdS) we can express the internal energy as du = TdS pdv By a transformation of variables we define H = U + PV or dh = d(u+pv) = du + PdV + VdP. Therefore, dh = TdS + VdP These two expressions are known as total derivatives.
The total derivative The total derivative permits more than one variable to change at one time. We can write the state functions U and H in the following form to make it clearer what the variable dependencies are: which implies that du = U S V ds + U V S dv dh = H S P ds + H P S dp T = U, P = U S V V S T = H, V = H S P P S
Definition of an exact differential In the general case we can write a total derivative as: df = F x dx + F y dy or df = F dx + x F dy y y x If the total derivative is an exact differential then the second derivatives of both terms in the differential are equal. F yx = F x y = F y x = F x y = F y x = F xy
Second derivatives of the internal energy function The differential form of the internal energy is: du = TdS -PdV Using the definitions we determined above: T = U S V, P = U V S The second derivatives can be written as: T V = U V S = U S V = P S
Second derivatives of the enthalpy function The differential form of the enthalpy is: dh = TdS + VdP Using the definitions we determined above: T = H S P, V = H P S The second derivatives can be written as: T P = H P S = H S P = V S
Second derivatives of the Gibbs free energy function The differential form of the Gibbs energy is: dg = VdP - SdT Using the definitions we determined above: V = G P T, S = G T P The second derivatives can be written as: S P = G P T = G T P = V T
Second derivatives of the Helmholtz free energy function The differential form of the Gibbs energy is: da = - PdV - SdT Using the definitions we determined above: P = A V T, S = A T V The second derivatives can be written as: P T = A T V = A V T = S V
Thermodynamics Applications of Exact Differentials Heat Capacity of Liquids and Solids NC State University
Thermodynamic equation of state From the master equation for internal energy: We can take the derivative with respect to volume at constant temperature to obtain: We can use one of the Maxwell relations to obtain: The equation is known as a thermodynamic equation of state. It relates the internal energy dependence on volume only to temperature and pressure.
The significance of the internal pressure The calculation of the internal pressure is: We can see that this quantity is a correction to the ideal gas law that depends on intermolecular interactions described by the a coefficient in the van der Waal s equation of state.
Dependence of internal energy on changes in volume and temperature We have seen from the first law that the internal energy change in the system is equal to the work done and the heat transferred. Work depends on volume changes and heat transfer leads to changes in temperature. Thus, it is logical that the initial description of internal energy depends on volume and temperature. Thus, for example the dependence of U on volume at constant temperature is: U = U + U V T dv And similarly the dependence of U on temperature at constant volume is: U = U + U T V dt
Dependence of internal energy on changes in volume and temperature Putting these together we have: U = U + U V T dv + U T V dt which can be written as the total derivative: du = U V T dv + U T V dt This called the total derivative. The idea is that it includes the variables in a space. Here the space is V,T space. We shall see that these are not the only possible variables that can describe the dependence of U.
Dependence of internal energy on changes in volume and temperature The partial derivatives are slopes in the space. U V T = p V, U T V = C V We have already seen C V and we know it as the heat capacity At constant volume. However, p V is new. p V is the change in the internal energy when the volume of a substance is changed at constant temperature. If the intermolecular interactions are zero (ideal gas), then p V will be zero. However, real interactions between molecules can give rise to non-zero p V. Joules set out to measure p V in an expansion experiment, but was not successful.
Expansion coefficient and isothermal compressibility In order to proceed with the next level of analysis we introduce the expansion coefficient: = 1 V V T P The expansion coefficient is a measure of the change in molar volume (i.e. also inverse density) of a material with temperature. Many substances expand as the temperature is increased, hence the name, expansion coefficient. The isothermal compressibility is = 1 V V P T a measure of the degree to which a substance will have a higher density (smaller molar volume) at high pressure.
Heat capacity relationships The molar heat capacity at constant pressure and constant Volume are related by: The above expression applies only to an ideal gas. We have seen the utility of that expression in that it applies to monatomic, diatomic and polyatomic gasses. For liquids and solids we have the formula:
Heat capacity relationships To derive the expression for the heat capacity difference We return to the definition. Note that we will assume that All extensive quantities are molar quantiites in this derivation. Therefore
Heat capacity relationships We can also write this as We use Euler s chain relation, We find that which gives us the result
The Joules-Thompson coefficient The differential of the enthalpy can be expressed as: Where m is the Joules-Thomson coefficient. The enthalpy change be written: Therefore: dh = mc P dp + C P dt dh = H P T dp + H T P dt H = T H = mc P T P H T P P The derivation of the above expression relies on the permutation relation: T P H P H T H T P = 1
Modern measurement of the Joules-Thompson coefficient It is relatively difficult to perform measurements under constant H (isenthalpic or adiabatic) conditions. Therefore, the modern measurements are at constant temperature. In other words, H P T = m T is measured. Based on the expression on the previous slide: m T = T H = mc P H T P P