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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

Chapter 5: 5.18 (page 163)

Imagine that you drop a brick on the ground and it lands with a thud. Apparently the energy of this system tends to spontaneously decrease. Explain why.

Short Answer

Expert verified

The energy is transferred from the brick to the ground, so the energy of the system tends to spontaneously decrease.

Step by step solution

01

Given

A brick is dropped to the ground and it lands with a thud.

Energy of this system tends to spontaneously decrease, explain why?

02

Explanation

Helmholtz free energy can be determined by
F=U-T S
Where, F= Helmholtz free energy, U=Internal energy,T=absolute temperature of the system and S=entropy of the system.

The total energy of the brick = kinetic energy + potential energy

From the law of energy conservation this is always constant.

When the brick hits the ground, its kinetic energy is zero, but the potential energy remains the same. The kinetic energy is redistributed into the thermal energy of the molecules of the brick and to the ground where the brick hits.

So a part of the energy is transferred from the brick to the ground.

Therefore the energy of the system tends to spontaneously decrease.

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Most popular questions from this chapter

What happens when you add salt to the ice bath in an ice cream maker? How is it possible for the temperature to spontaneously drop below 0"C? Explain in as much detail as you can.

Write down the equilibrium condition for each of the following reactions:

$(a) 2 H \leftrightarrow H_{2} (b) 2 CO + O_{2} \leftrightarrow 2 {CO}_{2} (c) {CH}_{4} + 2 O_{2} \leftrightarrow 2 H_{2} O + {CO}_{2} (d) H_{2} {SO}_{4} \leftrightarrow 2 H^{+} + {SO}_{4}^{2 -} (e) 2 p + 2 n \leftrightarrow {}^{4}{He}$

In a hydrogen fuel cell, the steps of the chemical reaction are
$at - electrode: H_{2} + 2 {OH}^{-} ⟶ 2 H_{2} O + 2 e^{-} at + electrode: \frac{1}{2} O_{2} + H_{2} O + 2 e^{-} ⟶ 2 {OH}^{-}$

Calculate the voltage of the cell. What is the minimum voltage required for electrolysis of water? Explain briefly.

The first excited energy level of a hydrogen atom has an energy of 10.2 eV, if we take the ground-state energy to be zero. However, the first excited level is really four independent states, all with the same energy. We can therefore assign it an entropy of $S = k \ln 4$ , since for this given value of the energy, the multiplicity is 4. Question: For what temperatures is the Helmholtz free energy of a hydrogen atom in the first excited level positive, and for what temperatures is it negative? (Comment: When F for the level is negative, the atom will spontaneously go from the ground state into that level, since F=0 for the ground state and F always tends to decrease. However, for a system this small, the conclusion is only a probabilistic statement; random fluctuations will be very significant.)

The partial-derivative relations derived in Problems 1.46,3.33, and 5.12, plus a bit more partial-derivative trickery, can be used to derive a completely general relation between $C_{P}$ and $C_{V}$ .

(a) With the heat capacity expressions from Problem 3.33 in mind, first consider $S$ to be a function of $T$ and $V .$ Expand $d S$ in terms of the partial derivatives $(\partial S / \partial T)_{V}$ and $(\partial S / \partial V)_{T}$ . Note that one of these derivatives is related to $C_{V}$

(b) To bring in $C_{P}$ , considerlocalid="1648430264419" $V$ to be a function of $T$ and P and expand dV in terms of partial derivatives in a similar way. Plug this expression for dV into the result of part (a), then set $d P = 0$ and note that you have derived a nontrivial expression for $(\partial S / \partial T)_{P}$ . This derivative is related to $C_{P}$ , so you now have a formula for the difference $C_{P} - C_{V}$

(c) Write the remaining partial derivatives in terms of measurable quantities using a Maxwell relation and the result of Problem 1.46. Your final result should be

$C_{P} = C_{V} + \frac{T V β^{2}}{κ_{T}}$

(d) Check that this formula gives the correct value of $C_{P} - C_{V}$ for an ideal gas.

(e) Use this formula to argue that $C_{P}$ cannot be less than $C_{V}$ .

(f) Use the data in Problem 1.46 to evaluate $C_{P} - C_{V}$ for water and for mercury at room temperature. By what percentage do the two heat capacities differ?

(g) Figure 1.14 shows measured values of $C_{P}$ for three elemental solids, compared to predicted values of $C_{V}$ . It turns out that a graph of $β$ vs.T for a solid has same general appearance as a graph of heat capacity. Use this fact to explain why $C_{P}$ and $C_{V}$ agree at low temperatures but diverge in the way they do at higher temperatures.

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