Energy in Thermal Physics

Your 200 g cup of tea is boiling-hot. About how much ice should you add to bring it down to a comfortable sipping temperature of $65°\mathrm{C}$ ? (Assume that the ice is initially $65°\mathrm{C}$. The specific heat capacity of ice is  $0.5 \mathrm{cal}/\mathrm{g}°\mathrm{C}$.

 When spring finally arrives in the mountains, the snow pack may be two meters deep, composed $50\mathrm{\%}$ of ice and $50\mathrm{\%}$ of  air. Direct sunlight provides about $1000{\text{ watts/m }}^2$ to earth's surface, but the snow might reflect $90\mathrm{\%}$ of this energy. Estimate how many weeks the snow pack should last, if direct solar radiation is the only source of energy.

 Your $200 g$ cup of tea is boiling-hot. About how much ice should you add to bring it down to a comfortable sipping temperature of ${65}^{\circ }\mathrm{C}$ ? (Assume that the ice is initially at $-{15}^{\circ }\mathrm{C}$. The specific heat capacity of ice is $(\left.0.5\mathrm{cal}/\mathrm{g}{\cdot }^{\circ }\mathrm{C}.\right)$.)

Problem 1.49. Consider the combustion of one mole of ${\mathrm{H}}_2$ with$1 / 2$  mole of ${\mathrm{O}}_2$ under standard conditions, as discussed in the text. How much of the heat energy produced comes from a decrease in the internal energy of the system, and how much comes from work done by the collapsing atmosphere? (Treat the volume of the liquid water as negligible.)

 Consider the combustion of one mole of methane gas:

${\mathrm{CH}}_4\text{ (gas) }+2{\mathrm{O}}_2\text{ (gas) }⟶{\mathrm{CO}}_2\text{ (gas) }+2{\mathrm{H}}_2\mathrm{O}\left(\text{ gas }\right)$

The system is at standard temperature $(298 \mathrm{K})$ and pressure $\left({10}^5 \mathrm{Pa}\right)$ both before and after the reaction.

(a) First imagine the process of converting a mole of methane into its elemental constituents (graphite and hydrogen gas). Use the data at the back of this book to find $\Delta H$for this process.

(b) Now imagine forming a mole of ${\mathrm{CO}}_2$ and two moles of water vapor from their elemental constituents. Determine $\Delta H$ for this process.

(c) What is $\Delta H$for the actual reaction in which methane and oxygen form carbon dioxide and water vapor directly? Explain.

(d) How much heat is given off during this reaction, assuming that no "other" forms of work are done?

(e) What is the change in the system's energy during this reaction? How would your answer differ if the${\mathrm{H}}_2\mathrm{O}$ ended up as liquid water instead of vapor?

(f) The sun has a mass of$2\times {10}^{30} \mathrm{kg}$ and gives off energy at a rate of $3.9\times$ ${10}^{26}$ watts. If the source of the sun's energy were ordinary combustion of a chemical fuel such as methane, about how long could it last?

The enthalpy of combustion of a gallon ($3.8$ liters) of gasoline is about $31,000 kcal$. The enthalpy of combustion of an ounce $\left(28 g\right)$ of corn flakes is about$100 kcal$. Compare the cost of gasoline to the cost of corn flakes, per calorie.

Look up the enthalpy of formation of atomic hydrogen in the back of this book. This is the enthalpy change when a mole of atomic hydrogen is formed by dissociating $1/2$mole of molecular hydrogen (the more stable state of the element). From this number, determine the energy needed to dissociate a single $H_2$molecule, in electron-volts.

A $60-kg$hiker wishes to climb to the summit of Mt. Ogden, an ascent of$5000$ vertical feet$(1500 m)$ .
$\left(a\right)$ Assuming that she is $25\%$ efficient at converting chemical energy from food into mechanical work, and that essentially all the mechanical work is used to climb vertically, roughly how many bowls of corn flakes (standard serving size $1$ounce, $100$ kilocalories) should the hiker eat before setting out?
$\left(b\right)$ As the hiker climbs the mountain, three-quarters of the energy from the corn flakes is converted to thermal energy. If there were no way to dissipate this energy, by how many degrees would her body temperature increase?
$\left(c\right)$In fact, the extra energy does not warm the hiker's body significantly; instead, it goes (mostly) into evaporating water from her skin. How many liters of water should she drink during the hike to replace the lost fluids? (At${25}^{\circ }C$ , a reasonable temperature to assume, the latent heat of vaporization of water is $580cal/g$ more than at ${100}^{\circ }C$).

Heat capacities are normally positive, but there is an important class of exceptions: systems of particles held together by gravity, such as stars and star clusters.
$\left(a\right)$ Consider a system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is- $2$times the total kinetic energy.
$\left(b\right)$ The conclusion of part $\left(a\right)$ turns out to be true, at least on average, for any system of particles held together by mutual gravitational attraction:

${\overline{U}}_{\text{potential }}=-2{\overline{U}}_{\text{kinetic }}$

Here each $U$ refers to the total energy (of that type) for the entire system, averaged over some sufficiently long time period. This result is known as the virial theorem. (For a proof, see Carroll and Ostlie ($1996$), Section $2.4$.) Suppose, then, that you add some energy to such a system and then wait for the system to equilibrate. Does the average total kinetic energy increase or decrease? Explain.

$\left(c\right)$ A star can be modeled as a gas of particles that interact with each other only gravitationally. According to the equipartition theorem, the average kinetic energy of the particles in such a star should be $\frac{3}{2}KT$, where $T$ is the average temperature. Express the total energy of a star in terms of its average temperature, and calculate the heat capacity. Note the sign.
$\left(d\right)$Use dimensional analysis to argue that a star of mass $M$ and radius $R$should have a total potential energy of $-G{M}^2/R$, times some constant of order $1.$
$\left(e\right)$Estimate the average temperature of the sun, whose mass is $2\times {10}^{30} kg$and whose radius is $7\times {10}^8 m$. Assume, for simplicity, that the sun is made entirely of protons and electrons.

Use the data at the back of this book to determine $\mathrm{\Delta H}$ for the combustion of a mole of glucose,

${\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6+6{\mathrm{O}}_2⟶6{\mathrm{CO}}_2+6{\mathrm{H}}_2\mathrm{O}.$

This is the (net) reaction that provides most of the energy needs in our bodies.

Home owners and builders discuss thermal conductivities in terms of the  value ($\mathbf{R}$ for resistance) of a material, defined as the thickness divided by the thermal conductivity:

$R\equiv \frac{\Delta x}{k_t}$

(a) Calculate the $R$ value of a $1/8\text{-inch }$($3.2\mathrm{mm}$) piece of plate glass, and then of a $1\mathrm{mm}$ layer of still air. Express both answers in SI units.

(b) In the United States, $R$ values of building materials are normally given in English units,${}^{\circ }\mathrm{F}\cdot {\mathrm{ft}}^2\cdot \mathrm{hr}/\mathrm{Btu}$. A Btu, or British thermal unit, is the energy needed to raise the temperature of a pound of water $1^{\circ }\mathrm{F}$. Work out the conversion factor between the SI and English units for values. Convert your answers from part (a) to English units.

(c) Prove that for a compound layer of two different materials sandwiched together (such as air and glass, or brick and wood), the effective total $R$ value is the sum of the individual $R$ values.

(d) Calculate the effective $R$ value of a single piece of plate glass with a $1.0$$\mathrm{mm}$ layer of still air on each side. (The effective thickness of the air layer will depend on how much wind is blowing; $1\mathrm{mm}$ is of the right order of magnitude under most conditions.) Using this effective $R$ value, make a revised estimate of the heat loss through a $1-{\mathrm{m}}^2$ single-pane window when the temperature in the room is ${20}^{\circ }\mathrm{C}$ higher than the outdoor temperature.

Calculate the rate of heat conduction through a layer of still air that is $1 mm$ thick, with an area of $1 m^2$, for a temperature difference of ${20}^{\circ }C$.

According to a standard reference table, the R value of a 3.5 inch-thick vertical air space (within a wall) is 1.0 R in English units), while the R value of a 3.5-inch thickness of fiberglass batting is 10.9. Calculate the R value of a 3.5-inch thickness of still air, then discuss whether these two numbers are reasonable. (Hint: These reference values include the effects of convection.)

Make a rough estimate of the total rate of conductive heat loss through the windows, walls, floor, and roof of a typical house in a cold climate. Then estimate the cost of replacing this lost energy over a month. If possible, compare your estimate to a real utility bill. (Utility companies measure electricity by the kilowatt-hour, a unit equal to  MJ. In the United States, natural gas is billed in terms, where 1 therm = 10⁵ Btu. Utility rates vary by region; I currently pay about 7 cents per kilowatt-hour for electricity and 50 cents per therm for natural gas.)

Geologists measure conductive heat flow out of the earth by drilling holes (a few hundred meters deep) and measuring the temperature as a function of depth. Suppose that in a certain location the temperature increases by ${20}^{\circ }\mathrm{C}$ per kilometer of depth and the thermal conductivity of the rock is $2.5\mathrm{W}/\mathrm{m}\cdot \mathrm{K}$. What is the rate of heat conduction per square meter in this location? Assuming that this value is typical of other locations over all of the earth's surface, at approximately what rate is the earth losing heat via conduction? (The radius of the earth is $6400\mathrm{km}$.)

A frying pan is quickly heated on the stovetop  ${200}^{\circ }\mathrm{C}$ It has an iron handle that is $20 cm$ long. Estimate how much time should pass before the end of the handle is too hot to grab with your bare hand. (Hint: The cross-sectional area of the handle doesn't matter. The density of iron is about$7.9\mathrm{g}/{\mathrm{cm}}^3$ and its specific heat is $0.45\mathrm{J}/\mathrm{g}{\cdot }^{\circ }\mathrm{C}$).

Consider a uniform rod of material whose temperature varies only along its length, in the $x$ direction. By considering the heat flowing from both directions into a small segment of length $\mathrm{\Delta }x$

derive the heat equation,

$\frac{\mathrm{\partial }T}{\mathrm{\partial }t}=K\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}$

where $K=k_t/c{\rho }_i$, $c$ is the specific heat of the material, and $\rho$ is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that $K$ is independent of temperature, show that a solution of the heat equation is

$T(x,t)=T_0+\frac{A}{\sqrt{t}}{e}^{-x^2/4Kt},$

where $T_0$ is a constant background temperature and $A$ is any constant. Sketch (or use a computer to plot) this solution as a function of $x$, for several values of $t$. Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.

At about what pressure would the mean free path of an air molecule at room temperature equal $10 cm$, the size of a typical laboratory apparatus?

Make a rough estimate of thermal conductivity of helium at room temperature. Discuss your result, explaining why it differs the value for air

Make a rough estimate of the thermal conductivity of helium at room temperature. Discuss your result, explaining why it differs from the value for air.

Pretend that you live in the $19$th century and don't know the value of Avogadro's number* (or of Boltzmann's constant or of the mass or size of any molecule). Show how you could make a rough estimate of Avogadro's number from a measurement of the thermal conductivity of gas, together with other measurements that are relatively easy.

In analogy with the thermal conductivity, derive an approximate formula for the viscosity of an ideal gas in terms of its density, mean free path, and average thermal speed. Show explicitly that the viscosity is independent of pressure and proportional to the square root of the temperature. Evaluate your formula numerically for air at room temperature and compare to the experimental value quoted in the text.

Make a rough estimate of how far food coloring (or sugar) will diffuse through water in one minute.

 Suppose you open a bottle of perfume at one end of a room. Very roughly, how much time would pass before a person at the other end of the room could smell the perfume, if diffusion were the only transport mechanism? Do you think diffusion is the dominant transport mechanism in this situation?

Consider a narrow pipe filled with fluid, where the concentration of a specific type of molecule varies only along its length (in the x direction). Fick's second law is derived by considering the flux of these particles from both directions into a short segment$∆x$

$\frac{\partial n}{\partial t}=D\frac{{\partial }^{2}n}{\partial x^2}$

 In analogy with the thermal conductivity, derive an approximate formula for the diffusion coefficient of an ideal gas in terms of the mean free path and the average thermal speed. Evaluate your formula numerically for air at room temperature and atmospheric pressure, and compare to the experimental value quoted in the text. How does D depend on T, at fixed pressure?