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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

Chapter 1: Q 1.5. (page 5)

When you're sick with a fever and you take your temperature with a thermometer, approximately what is the relaxation time?

Short Answer

Expert verified

1 minute.

Step by step solution

01

Given information

One sick person measures the temperature of the body and try to fin out the relaxation time.

02

Calculation

Two objects take time to attain thermal equilibrium. The relaxation time is defined as the time taken by an object to achieve the temperature which is $(\frac{1}{e})$ times of its initial value.

Usually when one takes the temperature by using a thermometer, one should wait around a minute to attain equilibrium & so the relaxation time will be approximate 1 minute.

03

Final answer

The relaxation time will be approximately 1 minute

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Most popular questions from this chapter

Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume. To see why, estimate the pressure needed to keep $V$ fixed as $T$ increases, as follows.

(a) First imagine slightly increasing the temperature of a material at constant pressure. Write the change in volume, $d V_{1}$ , in terms of $d T$ and the thermal expansion coefficient $β$ introduced in Problem 1.7.

(b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume for this process, $d V_{2}$ , in terms of $d P$ and the isothermal compressibility $κ_{T}$ , defined as

$κ_{T} \equiv - \frac{1}{V} {(\frac{\partial V}{\partial P})}_{T}$

(c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Then the ratio of $d P to d T$ is equal to $(\partial P / \partial T)_{V}$ , since there is no net change in volume. Express this partial derivative in terms of $β and κ_{T}$ . Then express it more abstractly in terms of the partial derivatives used to define $β and κ_{T}$ . For the second expression you should obtain

${(\frac{\partial P}{\partial T})}_{V} = - \frac{(\partial V / \partial T)_{P}}{(\partial V / \partial P)_{T}}$

This result is actually a purely mathematical relation, true for any three quantities that are related in such a way that any two determine the third.

(d) Compute $β, κ_{T}, and (\partial P / \partial T)_{V}$ for an ideal gas, and check that the three expressions satisfy the identity you found in part (c).

(e) For water at $25^{\circ} C, β = 2.57 \times 10^{- 4} K^{- 1} and κ_{T} = 4.52 \times 10^{- 10} {Pa}^{- 1}$ . Suppose you increase the temperature of some water from $20^{\circ} C to 30^{\circ} C$ . How much pressure must you apply to prevent it from expanding? Repeat the calculation for mercury, for which $(at 25^{\circ} C) β = 1.81 \times 10^{- 4} K^{- 1}$ and $κ_{T} = 4.04 \times 10^{- 11} {Pa}^{- 1}$

Given the choice, would you rather measure the heat capacities of these substances at constant $v$ or at constant $p$ ?

Even at low density, real gases don’t quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial

expansion,

$P V - n R T (1 + \frac{B (T)}{(V / n)} + \frac{C (T)}{{(V / n)}^{2}} + \dots)$

where the functions $B (T)$ , $C (T)$ , and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations, it’s sufficient to omit the third term and concentrate on the second, whose coefficient $B (T)$ is called the second virial coefficient (the first coefficient is 1). Here are some measured values of the second virial coefficient for nitrogen ( $N_{2}$ ):

$T (K)$	$B (c m^{3} / m o l)$
100	–160
200	–35
300	–4.2
400	9.0
500	16.9
600	21.3

For each temperature in the table, compute the second term in the virial equation, $B (T) / (V / n)$ , for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions.
Think about the forces between molecules, and explain why we might expect $B (T)$ to be negative at low temperatures but positive at high temperatures.
Any proposed relation between $P$ , $V$ , and $T$ , like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation,
$(P + \frac{a n^{2}}{V^{2}}) (V - n b) = n R T$
where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients ( $B$ and $C$ ) for a gas obeying the van der Waals equation, in terms of $a$ and $b$ . (Hint: The binomial expansion says that ${(1 + x)}^{p} \approx 1 + p x + \frac{1}{2} p (p - 1) x^{2}$ , provided that $| p x | ≪ 1$ . Apply this approximation to the quantity ${[1 - (n b / V)]}^{- 1}$ .)
Plot a graph of the van der Waals prediction for $B (T)$ , choosing $a$ and $b$ so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)

Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume.

Sketch a graph of pressure vs. volume for this process.
Calculate the work done on the gas during this process, assuming that there are no “other” types of work being done.
Calculate the change in the helium’s energy content during this process.
Calculate the amount of heat added to or removed from the helium during this process.
Describe what you might do to cause the pressure to rise as the helium expands.

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

$R \equiv \frac{Δ x}{k_{t}}$

(a) Calculate the $R$ value of a $1 / 8 -inch$ ( $3.2 mm$ ) piece of plate glass, and then of a $1 mm$ 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, $^{\circ} F \cdot {ft}^{2} \cdot hr / Btu$ . A Btu, or British thermal unit, is the energy needed to raise the temperature of a pound of water $1^{\circ} F$ . 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 $1.0$ $mm$ layer of still air on each side. (The effective thickness of the air layer will depend on how much wind is blowing; $1 mm$ 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 $1 - m^{2}$ single-pane window when the temperature in the room is $20^{\circ} C$ higher than the outdoor temperature.

In Problem 1.16 you calculated the pressure of earth's atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottom most 10-15 km of the atmosphere (called the troposphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient |dT/dz| exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling.
(a) Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation
$\frac{d T}{d P} = \frac{2}{f + 2} \frac{T}{P}$
(b) Assume that dT/dz is just at the critical value for convection to begin, so that the vertical forces on a convecting air mass are always approximately in balance. Use the result of Problem 1.16b to find a formula for dT/dz in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately -10^oC/ km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.

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