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Chapter 1: Q.1.43 (page 31)

Calculate the heat capacity of liquid water per molecule, in terms of K . Suppose (incorrectly) that all the thermal energy of water is stored in quadratic degrees of freedom. How many degrees of freedom would each molecule have to have?

Short Answer

Expert verified

Short Answer :

Heat capacity per molecule Cm=9.072k

Degrees of freedom f = 18

Step by step solution

01

Given Information :

Water's thermal energy is stored in quadratic degrees of freedom

02

Explanation 

The specific heat capacity of water is cwater=1cal路g-1路C-1=4.186J-1g-1路C-1, and the molar weight of water is 18.015g路mol -1so in 1gramof water.

N=Number of molecules in one moleMass of one mole=6.022脳102318.015

N=3.343脳1022molecules

The heat capacity of a single molecule is therefore:

Cm=CN=4.1863.343脳1022=1.252脳10-22J路C-1Cm=1.252脳10-22脳kk=1.252脳10-22脳k1.38脳10-23Cm=9.072k

03

Explanation

The heat capacity is linearly proportional to the number of degrees of freedom, according to the equipartition theorem,

Cm=fk2

Therefore now on comparing heat capacities, we have

fk2=9.072k

f=18

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Most popular questions from this chapter

Even at low density, real gases don鈥檛 quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial

expansion,

PV鈭�nRT(1+B(T)(V/n)+C(T)(V/n)2+鈰�)

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鈥檚 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 (N2):

T(K)
B(cm3/mol)
100鈥�160
200鈥�35
300鈥�4.2
4009.0
50016.9
60021.3
  1. 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.
  2. 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.
  3. Any proposed relation between P, V, andT, 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+an2V2)(V鈭�nb)=nRT
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  4. Plot a graph of the van der Waals prediction for B(T), choosing aand bso 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.)

The Rankine temperature scale(abbreviated 掳R) uses the same scale size degrees as Fahrenheit, but measured up from absolute zero like Kelvin(so Rankine is to Fahrenheit as Kelvin is to Celsius). Find the conversion formula between Rankine and Fahrenheit and also between Rankine and Kelvin. What is the room temperature on the Rankine scale?

Consider a uniform rod of material whose temperature varies only along its length, in the xdirection. By considering the heat flowing from both directions into a small segment of length 螖x

derive the heat equation,

鈭�T鈭�t=K鈭�2T鈭�x2

where K=kt/c蚁i, cis the specific heat of the material, and 蚁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 Kis independent of temperature, show that a solution of the heat equation is

T(x,t)=T0+Ate鈭�x2/4Kt,

where T0is a constant background temperature and Ais 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.

Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecules are moving faster, on average? By what factor?

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
dTdP=2f+2TP
(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 -10oC/ km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.
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