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Chapter 1: Q.1.23 (page 17)

Calculate the total thermal energy in a liter of helium at room temperature and atmospheric pressure. Then repeat the calculation for a liter of air.

Short Answer

Expert verified

The total thermal energy in a liter is for heliumU=151.987J and for airU=253.312J.

Step by step solution

01

Step 1. Given data.

A liter of helium at ambient temperature at the pressure of one atmosphere1atm=101325Pa.

02

Step 2. Explanation.

Whereas a monatomic gas, like helium, has only three translational degrees of freedom, it is the simplest instance.

U=f2NkT=32NkT

Thermal energy,

U=32NkT=32PV=32×101325×1×10-3=151.987J

03

Step 3. Total thermal energy.

A rotation around the bond's axis isn't counted because it doesn't change the molecule's position.

f=5

U=f2NkT=52NkTU=52PV=52×101325×1×10-3=253.312.J

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Most popular questions from this chapter

Derive the equation 1.40 from the equation 1.39

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,

PV−nRT(1+B(T)(V/n)+C(T)(V/n)2+⋯)

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 (N2):

T(K)
B(cm3/mol)
100–160
200–35
300–4.2
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−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≈1+px+12p(p−1)x2, provided that |px|≪1. Apply this approximation to the quantity [1−(nb/V)]−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.)

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.

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?

By applying a pressure of 200atm, you can compress water to 99%of its usual volume. Sketch this process (not necessarily to scale) on aPV diagram, and estimate the work required to compress a liter of water by this amount. Does the result surprise you?
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