Agree that the derivative of entropy w/r to P at constant T is the negative of...

Agree that the derivative of entropy w/r to P at constant T is the negative of the product of volume and coefficient of thermal expansion? And that therefore deltaS is negligible for liquids and solids but can be quite substantial for gases? Please explain it!

Expert Solution

The temperature dependence of volume under constant pressure can be transformed to the pressure dependence of entropy under constant temperature through the Maxwell relation, resulting in the volume thermal expansion coefficient, α_V, expressed as follows

where V is the volume, T the temperature, P the pressure, G the Gibbs free energy and S the entropy. This equation introduces quantitative understanding of thermal expansion in terms of entropy. Therefore, the entropy of materials can either decrease or increase with pressure even though it always increases with temperature in a stable system due to the stability requirements. dU = TdS + PdV, describes the combined effects of the first and second Laws of Thermodynamics. The first term of the equation deals primarily with the energy associated with phase changes, where the entropy of a substance in a given volume changes. The second term deals primarily with the energy associated with volume changes in gas, where the temperature and entropy per particle remain constant. So looking at the equation, the total amount of heat energy in a substance must increase if it changes phase from solid to liquid, solid to gas, or liquid to gas. Conversely, the heat energy must decrease if it changes phase from liquid to solid, gas to liquid, or gas to solid. For gasses, the second term will be very large compared to liquid and solid.

genius_generous answered 6 months ago

Given is a population of wolves (W) and rabbits (R). R[t+1] = R[t]+ gR[t] (1...

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Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker...

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Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x...

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Suppose S = {p, q, r, s, t, u} and A = {p, q, s, t}...

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Let T : Pn → R be defined by T(p(x)) = the sum of all the...

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The following are at constant T and P. The conditions described in (a) and (b) below...

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Let r : R->R3 be a path with constant speed, satisfying r"(t) = (4 sin(2t))i +...

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P(W)= 0.3 P(low/W) = 0.50 P(low/S) = 0.10 P(S)= 0.7 P(medium/W)= 0.40 P(medium/S)= 0.25 P(high/W) =...

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Suppose that X ~ NB(r, p) [negative binomial distribution] and that Y ~ B(n, p) [binomial]....

Suppose that X ~ NB(r, p) [negative binomial distribution] and that Y ~ B(n, p) [binomial]. a. Give a probabilistic argument to show that P(X > n) = P(Y < r). b. Use the FPF to express the equality in part (a) in terms of PMFs. c. Using the complementation rule, how many terms of the PMF of X must be evaluated to determine P(X > n)? d. How many terms of the PMF of Y must be evaluated to...

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