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Chapter 5: Q 5.82 (page 208)

Use the result of the previous problem to calculate the freezing temperature of seawater.

Short Answer

Expert verified

Therefore, the freezing temperature of seawater is-2.167°C

Step by step solution

01

Given information

From the previous problem we have:

T-T0=-NBkT02L

02

Explanation

from the previous problem, we have

T-T0=-NBkT02L

but,NBk=nBRr, so

T-T0=-nBRT02L

Where,

Tois the freezing temperature of water

nBis the number of moles of salt

Lis the fusion latent energy of water and is given as L=333.7×103J

03

Calculations

We have 35 grammes of salt in one kilogram of water because sea water contains calcium and sodium, which have an average molar mass of 30 g/mol, hence in 35g the number of moles are

nB=35g30g/mol=1.167mol

Substitute the values,

role="math" localid="1647204325114" T-T0=-(1.167mol)(8.314J/mol·K)(273K)2333.7×103J=-2.167KT=273K-2.167K=270.833KT=-2.167°C

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Most popular questions from this chapter

Consider the production of ammonia from nitrogen and hydrogen,

N2+3H2→2NH3

at 298 K and 1 bar. From the values of ΔH and S tabulated at the back of this book, compute ΔG for this reaction and check that it is consistent with the value given in the table.

Use a Maxwell relation from the previous problem and the third law of thermodynamics to prove that the thermal expansion coefficient (defined in Problem 1.7) must be zero at T=0.

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%?

Functions encountered in physics are generally well enough behaved that their mixed partial derivatives do not depend on which derivative is taken first. Therefore, for instance,

∂∂V∂U∂S=∂∂S∂U∂V

where each ∂/∂Vis taken with Sfixed, each ∂/∂Sis taken with Vfixed, and Nis always held fixed. From the thermodynamic identity (forU) you can evaluate the partial derivatives in parentheses to obtain

∂T∂VS=-∂P∂SV

a nontrivial identity called a Maxwell relation. Go through the derivation of this relation step by step. Then derive an analogous Maxwell relation from each of the other three thermodynamic identities discussed in the text (for H,F,andG ). Hold N fixed in all the partial derivatives; other Maxwell relations can be derived by considering partial derivatives with respect to N, but after you've done four of them the novelty begins to wear off. For applications of these Maxwell relations, see the next four problems.

By subtracting μNfrom localid="1648229964064" U,H,F,orG,one can obtain four new thermodynamic potentials. Of the four, the most useful is the grand free energy (or grand potential),

Φ≡U-TS-μN.

(a) Derive the thermodynamic identity for Φ, and the related formulas for the partial derivatives ofΦwith respect toT,V, and μN

(b) Prove that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), Φtends to decrease.

(c) Prove thatϕ=-PV.

(d) As a simple application, let the system be a single proton, which can be "occupied" either by a single electron (making a hydrogen atom, with energy -13.6eV) or by none (with energy zero). Neglect the excited states of the atom and the two spin states of the electron, so that both the occupied and unoccupied states of the proton have zero entropy. Suppose that this proton is in the atmosphere of the sun, a reservoir with a temperature of 5800Kand an electron concentration of about 2×1019per cubic meter. Calculate Φfor both the occupied and unoccupied states, to determine which is more stable under these conditions. To compute the chemical potential of the electrons, treat them as an ideal gas. At about what temperature would the occupied and unoccupied states be equally stable, for this value of the electron concentration? (As in Problem 5.20, the prediction for such a small system is only a probabilistic one.)
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