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Chapter 1: Q.1.24 (page 17)

Calculate the total thermal energy in a gram of lead at room temperature, assuming that none of the degrees of freedom are "frozen out" (this happens to be a good assumption in this case).

Short Answer

Expert verified

The total thermal energyU=35.3J.

Step by step solution

01

Step 1. Introduction.

Each atom in lead has six degrees of freedom in terms of vibration (3kinetic and 3potential). There are no transitional degrees of freedom, and there are no rotational degrees of freedom for a single atom because the atoms are fixed in their locations in the lattice.

02

Step 2. Explanation.

Given the molar mass of lead, the number of molecules in one gram of lead is,

Hereu=1.66脳10kg27.

N=massmolar mass=1脳103kg207.2脳1.66脳1027kg=2.91脳1021

The thermal energy at T=293K

U=f2NkT=62脳2.91脳1021脳1.38脳1023脳293=35.3J

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Most popular questions from this chapter

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

Estimate the average temperature of the air inside a hot-air balloon (see Figure 1.1). Assume that the total mass of the unfilled balloon and payload is 500 kg. What is the mass of the air inside the balloon?

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
    where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (Band C) for a gas obeying the van der Waals equation, in terms of aand b. (Hint: The binomial expansion says that (1+x)p鈮�1+px+12p(p鈭�1)x2, provided that |px|鈮�1. Apply this approximation to the quantity [1鈭�(nb/V)]鈭�1.)
  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.)

Calculate the rate of heat conduction through a layer of still air that is1mmthick, with an area of 1m2, for a temperature difference of 20鈭�C.

During a hailstorm, hailstones with an average mass of 2g and a speed of 15 m/s strike a window pane at a 45o angle. The area of the window is 0.5 m2 and the hailstones hit it at a rate of 30 per second. What average pressure do they exert on the window? How does this compare to the pressure of the atmosphere?
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