The formula for derived in the previous problem can also be derived starting with the definitions of these quantities in terms of U and H. Do so. Most of the derivation is very similar, but at one point you need to use the relation .
The specific heat of a substance can be of two types:(i) specific heat at constant pressure CP(ii) specific heat at constant volume CV
They are given by
Where, U = internal energy, H = enthalpy, V = volume and P = pressure.
Lets consider U=U(V, T), and differentiate above expression with respect to T, we get
Similarly write expression for enthalpy H=U+P V.
Write the expression for d H at constant pressure d H=d U+P d V
Simplify, divide both sides of the above expression by d T,
Substitute in equation (2)
Now substitute TS for (U-F)
So, the value of
Ordinarily, the partial pressure of water vapour in the air is less than the equilibrium vapour pressure at the ambient temperature; this is why a cup of water will spontaneously evaporate. The ratio of the partial pressure of water vapour to the equilibrium vapour pressure is called the relative humidity. When the relative humidity is 100%, so that water vapour in the atmosphere would be in diffusive equilibrium with a cup of liquid water, we say that the air is saturated. The dew point is the temperature at which the relative humidity would be 100%, for a given partial pressure of water vapour.
(a) Use the vapour pressure equation (Problem 5.35) and the data in Figure 5.11 to plot a graph of the vapour pressure of water from 0°C to 40°C. Notice that the vapour pressure approximately doubles for every 10° increase in temperature.
(b) Suppose that the temperature on a certain summer day is 30° C. What is the dew point if the relative humidity is 90%? What if the relative humidity is 40%?
Consider an ideal mixture of just 100 molecules, varying in com- position from pure A to pure B. Use a computer to calculate the mixing entropy as a function of NA, and plot this function (in units of k). Suppose you start with all A and then convert one molecule to type B; by how much does the entropy increase? By how much does the entropy increase when you convert a second molecule, and then a third, from A to B? Discuss.
Repeat the previous problem for the diagram in Figure 5.35 (right), which has an important qualitative difference. In this phase diagram, you should find that and liquid are in equilibrium only at temperatures below the point where the liquid is in equilibrium with infinitesimal amounts of and . This point is called a peritectic point. Examples of systems with this behaviour include water + NaCl and leucite + quartz.
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