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Chapter 1: Q.1.28 (page 20)

Estimate how long it should take to bring a cup of water to boiling temperature in a typical 600-wattmicrowave oven, assuming that all the energy ends up in the water. (Assume any reasonable initial temperature for the water.) Explain why no heat is involved in this process.

Short Answer

Expert verified

Time is taken to boil2.616 minutes to raise the 20mlwater fromlocalid="1648467354265" 10掳Cto100掳C.

Step by step solution

01

Step 1. Finding Amount of heat.

Consider a mug filled with250mLcold10掳Cwater in a 600wattmicrowave. To achieve the boiling point, So must raise the temperature by90藲C.

Q=尘颁螖罢

m=Water mass.

C=Specific heat C=4.186J/g路C鈭�

螖罢=90掳Temp

Therefore,

Q=250脳4.186脳90=94185J.

02

Step 2. Boiling time.

The Power

P=Qt鈫�t=QP

Insert Qand Pvalues.

t=94185600=156.975s=2.616min.

Normally, heat is transferred from the hotter to the colder object. However, because there is no hot item in our scenario, the heat is transported to the mug via an electromagnetic wave generated by the magnetron.

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Most popular questions from this chapter

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鈭�x

鈭�n鈭�t=D鈭�2n鈭�x2

In a Diesel engine, atmospheric air is quickly compressed to about 1/20 of its original volume. Estimate the temperature of the air after compression, and explain why a Diesel engine does not require spark plugs.

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 Vfixed as Tincreases, as follows.

(a) First imagine slightly increasing the temperature of a material at constant pressure. Write the change in volume,dV1, in terms of dTand the thermal expansion coefficient 尾introduced in Problem 1.7.

(b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume for this process, dV2, in terms of dPand the isothermal compressibility 魏T, defined as

魏T鈮�鈭�1V鈭�V鈭�PT

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

鈭�P鈭�TV=鈭�(鈭�V/鈭�T)P(鈭�V/鈭�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 尾,魏T,and(鈭�P/鈭�T)Vfor an ideal gas, and check that the three expressions satisfy the identity you found in part (c).

(e) For water at 25鈭�C,尾=2.57脳10鈭�4K鈭�1and魏T=4.52脳10鈭�10Pa鈭�1. Suppose you increase the temperature of some water from 20鈭�Cto30鈭�C. How much pressure must you apply to prevent it from expanding? Repeat the calculation for mercury, for which (at25鈭�C)尾=1.81脳10鈭�4K鈭�1and魏T=4.04脳10鈭�11Pa鈭�1

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

Pretend that you live in the 19th 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.

By applying Newton鈥檚 laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by

cs=B蚁,

where 蚁is the density of the medium (mass per unit volume) and B is the bulk modulus, a measure of the medium鈥檚 stiffness? More precisely, if we imagine applying an increase in pressure 螖Pto a chunk of the material, and this increase results in a (negative) change in volume 螖V, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:

B=螖P鈥�螖V/V

This definition is still ambiguous, however, because I haven't said whether the compression is to take place isothermally or adiabatically (or in some other way).

  1. Compute the bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions.
  2. Argue that for purposes of computing the speed of a sound wave, the adiabatic B is the one we should use.
  3. Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the RMS speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature.
  4. When Scotland鈥檚 Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?
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