Ch 20: The Micro/Macro Connection

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 20: The Micro/Macro Connection

Problem 61a

Chapter 20, Problem 61a

What is the total rotational kinetic energy of 1.0 mol of nitrogen gas at 300 K?

Verified step by step guidance

Understand the problem: Rotational kinetic energy for a diatomic gas like nitrogen (N₂) is determined using the formula for the average rotational kinetic energy per molecule, which is \( KE_{rot} = \frac{1}{2} k_B T \) per degree of freedom. Since nitrogen has two rotational degrees of freedom, the total rotational kinetic energy per molecule is \( KE_{rot,total} = k_B T \).

Determine the total rotational kinetic energy for 1 mole of nitrogen gas. Since 1 mole contains \( N_A \) (Avogadro's number) molecules, the total rotational kinetic energy is given by \( KE_{rot,total} = N_A \cdot k_B \cdot T \).

Substitute the known constants into the formula: \( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \), \( k_B = 1.38 \times 10^{-23} \, \text{J/K} \), and \( T = 300 \, \text{K} \).

Perform the multiplication: Multiply \( N_A \), \( k_B \), and \( T \) to calculate the total rotational kinetic energy for 1 mole of nitrogen gas. Ensure the units are consistent throughout the calculation.

Interpret the result: The final value represents the total rotational kinetic energy of 1 mole of nitrogen gas at 300 K, expressed in joules (J).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation. It is given by the formula KE_rot = (1/2) I ω², where I is the moment of inertia and ω is the angular velocity. For gases, this energy is related to the degrees of freedom available for molecular rotation, which contributes to the overall kinetic energy of the gas.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding the behavior of gases under various conditions. In this context, it helps determine the total energy of the gas by linking temperature to kinetic energy, as temperature is a measure of the average kinetic energy of the gas molecules.

Degrees of Freedom

Degrees of freedom refer to the number of independent ways in which a system can move or store energy. For a diatomic gas like nitrogen, there are translational, rotational, and vibrational degrees of freedom. At room temperature, nitrogen primarily exhibits translational and rotational motion, with each degree of freedom contributing to the total kinetic energy, which can be calculated using the equipartition theorem.

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A water molecule has its three atoms arranged in a 'V' shape, so it has rotational kinetic energy around any of three mutually perpendicular axes. However, like diatomic molecules, its vibrational modes are not active at temperatures below 1000 K. What is the thermal energy of 2.0 mol of steam at a temperature of 160°C?

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Textbook Question

A gas of 1.0 x 10²⁰ atoms or molecules has 1.0 J of thermal energy. Its molar specific heat at constant pressure is 20.8 J/ mol K. What is the temperature of the gas?

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Textbook Question

A nitrogen molecule consists of two nitrogen atoms separated by 0.11 nm, the bond length. Treat the molecule as a rotating dumbbell and find the rms angular velocity at this temperature of a nitrogen molecule around the z-axis, as shown in Figure 20.10.

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A monatomic gas is adiabatically compressed to 1/8 of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not? The mean free path.

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2.0 g of helium at an initial temperature of 300 K interacts thermally with 8.0 g of oxygen at an initial temperature of 600 K. How much heat energy is transferred, and in which direction?

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A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. How much heat energy is transferred, and in which direction?

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